GRAU DE MATEMÀTIQUES
Treball final de grau
DYNAMICS OF FINITE BLASCHKE
PRODUCTS
Autor: Joan Núñez Corbacho
Director: Dra. Núria Fagella Rabionet
Realitzat a: Departament de Matemàtiques i Informàtica
Barcelona, 17 de gener de 2024
Acknowledgements
Before venturing into complex dynamics, I would like to thank all of those who made this
project possible. Hence, I want to thank my family and friends, who supported me throughout
these last six years and helped to make every single day enjoyable, even days fully spent in the
library, which were not few. I hope these words are enough to thank you for everything you
have given me throughout these years.
Nevertheless, I am certain I would never be able to find enough words to thank both my
advisers, Núria and Anna. I feel rewriting these fifty pages from scratch and filling them with
acknowledgements would not be enough, since it would not do justice to the help you have
given me, even during Christmas.
Abstract
The aim of this project is to characterise the dynamics of finite Blaschke products, which
are precisely the proper maps of the unit disk. It is proven that, inside the unit disk, all points
converge to a unique point, the Wolff-Denjoy point. We build a classification of finite Blaschke
products according to the position of the Wolff-Denjoy point and the dynamics around it.
Finally, we study the restriction of finite Blaschke products to the unit circle and calculate
explicitly a conjugacy to zd. We end this work by showing a brief example of generalised
Blaschke products, a nuanced variation of the previous family that presents rich dynamical
phenomena, such as the emergence of Herman rings.
2020 Mathematics Subject Classification. 30D45, 30J10, 37B10
Contents
Introduction iv
1 Holomorphic self-maps of the unit disk 1
1.1 Biholomorphisms of the unit disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Proper self-maps of the unit disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Derivatives of finite Blaschke products in the unit circle . . . . . . . . . . . . . . 6
1.4 Generalisations of finite Blaschke products . . . . . . . . . . . . . . . . . . . . . . 7
2 Complex dynamics 10
2.1 The dynamic partition of the Riemann Sphere: Fatou and Julia sets . . . . . . . . 10
2.2 Local theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Attracting and repelling case . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Super-attracting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Parabolic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Global theory: Structure of the Fatou set . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Iteration in the unit disk and the unit circle . . . . . . . . . . . . . . . . . . . . . . 22
3 Dynamics of finite Blaschke products 24
4 Dynamics of finite Blaschke products in the unit circle 30
4.1 Abstract conjugacy in the unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Study of Shub’s conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.1 Symbolic dynamics in the unit circle . . . . . . . . . . . . . . . . . . . . . . 34
4.2.2 Finding the conjugacy explicitly . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Chaos in finite Blaschke products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Dynamics of Generalised Blaschke products: an example 46
5.1 Existence of rotation domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.1.1 Conjugating to an irrational rotation . . . . . . . . . . . . . . . . . . . . . . 47
5.1.2 Finding a Diophantine rotation number . . . . . . . . . . . . . . . . . . . . 47
5.1.3 Extending the conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Herman rings in the family of degree 3 GFBP . . . . . . . . . . . . . . . . . . . . 49
ii
A Appendix: Tools from complex analysis 52
A.1 Basic concepts from complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . 52
A.2 The Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A.3 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.4 Uniformization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A.5 Proper maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Bibliography 56
Introduction
Complex dynamics comprises the study of the iterates of holomorphic maps on Riemann
surfaces. In other words, given a Riemann surface U and a holomorphic map f : U −→ U, our
focus lies on the sequence
z, f (z), f 2(z), . . .
for z ∈ U, where f n = f ◦ (n)· · · ◦ f denotes the n-th iterate of f at z. This string of values is
known as the orbit of z, and our ultimate goal is to understand the orbit of all points in U.
The first remarkable results in this field are relatively recent, from the beginning of the 20th
century. In fact, prior to the 1910s, very little was written on the topic, mainly centred in the
local behaviour around fixed points, aligning more with the field of functional equations rather
than complex dynamics. The initial spark can be traced back to Pierre Fatou in 1906, with the
study of rational maps with only one fixed point. With additional hypotheses, he showed that
orbits for these functions converge to this unique fixed point everywhere in the complex plane
except for a closed, perfect set.
Figure 1: An example of the earlier results obtained by Fatou in 1906. All points in the plane tend to the fixed
point ∞, except for a closed, perfect (and in this case, totally disconnected) set. Red regions indicate convergence
to ∞, and the yellow gradient indicates rate of convergence.
The first, more general results establishing the foundations of holomorphic dynamics came
a few years later, spurred by the Great Prize Award in Mathematics of 1918 from the Paris
Academy of Sciences. Motivated by this event, P. Fatou and Gaston Julia independently laid
the groundwork for complex dynamics, working in the iteration of rational maps in the Rie-
mann sphere, Cˆ = C ∪ {∞}. Their cornerstone was the concept of normality, introduced by
Introduction v
Paul Montel in 1911. Making use of this powerful technique, they divided the Riemann sphere
into two completely invariant sets, now bearing their names, the Fatou set and the Julia set.
In essence, the Fatou set contains the points with stable behaviour under iteration, in the sense
that every point has a neighbourhood in which the asymptotic behaviour of the orbit is the
same. On the other hand, the Julia set is the complement of the Fatou set, with chaotic dynam-
ics, where nearby points may present different orbits.
Consequently, the study of rational maps involves characterising these two sets and the
behaviour of the orbits within them. As Fatou already hinted in his earlier work, the Julia
set is closed and perfect. Thus, the Fatou set is open and composed of different connected
components, the Fatou components. This exploration continued for over half a century, when
Dennis Sullivan, in 1985, proved the No Wandering Domain Theorem (Theorem 2.28), asserting
that the orbits of Fatou components of a rational map must be eventually periodic. The pinnacle
of this discussion came with the Classification Theorem (Theorem 2.30), providing the ultimate
classification of these Fatou components.
Fatou and Julia’s pioneering work with rational maps in the Riemann sphere outlined the
field of complex dynamics. Today, however, holomorphic dynamics reaches its true extent with
the study of more general holomorphic maps in more varied Riemann surfaces. In fact, due
to the well-known Uniformization Theorem (Theorem A.12), the study of the dynamics of any
holomorphic function in a given Riemann surface is covered by the analysis of the iteration in
the three following cases: in the Riemann sphere, in the complex plane, and in the unit disk.
Thus, rational iteration represents only one of these three possibilities. We shall now move our
attention to the case of iteration in the unit disk,
D = {z ∈ C : |z| < 1} .
Due to earlier results by Riemann, Poincaré and Koebe, this last case encompasses iteration
on all simply connected Riemann surfaces whose boundary contains at least three points (see
Riemann Mapping Theorem A.10).
If the precursor of rational iteration is the study of functional equations, then the bedrock of
iteration in the unit disk is Schwarz Lemma from 1880 (Lemma A.1), and its later generalisation
by Carathéodory from 1912. In this case, the climax arrived in 1926, when Julius Wolff and
Arnaud Denjoy independently formulated the Wolff-Denjoy Theorem (Theorem 2.31), a result
strong enough to characterise every holomorphic self-map ofD. It asserts that if a holomorphic
map defined in the unit disk is not an automorphism, then there exists a unique point inD, the
Wolff-Denjoy point, towards which all orbits converge. Notably, the function is only required
to be holomorphic in D, with no assumption of extension to ∂D.
Now, with this context in mind, we shall move into the matter of study of this project, a
topic shaped by the two pinnacle results of complex dynamics, the Classification Theorem and
the Wolff-Denjoy Theorem: finite Blaschke products.
vi Introduction
Finite Blaschke products
Finite Blaschke products of degree k ≥ 1 are maps of the form
B(z) = eiθ
k
∏
l=1
z− wl
1− wlz , for θ ∈ R, wl ∈ D, z ∈ D.
They are named after Wilhelm Blaschke, who introduced them in 1915 in the limit case where k
approaches ∞, where they hold significant relevance in the context of Hardy spaces and inner
functions. Nevertheless, our focus in this work lies in their finite form, considering a finite
degree k ≥ 1. They are not only rational maps of degree k but also self-maps of the unit disk,
since, as one may notice from their definition, they are products of automorphisms of D. From
this duality stems the interesting dynamics of Blaschke products: they conform the intersection
between iteration in the Riemann sphere and iteration in the unit disk. And here lies the goal
of our work: to combine the results of Fatou and Julia with the results of Wolff and Denjoy, in
order to give an accurate description of the dynamics of this family of maps.
∞
CCˆ
0
(x, y,−z)
z
zR
(x, y, z)
Figure 2: Sketch of the reflection property of finite Blaschke products. Each point z ∈ D has a reflection point
zR = 1/z, and the extension of finite Blaschke products satisfies B(z)B(zR) = 1. Interestingly, considering the
stereographic projection, in the Riemann sphere Cˆ these points are reflections on the z-plane.
Before venturing into their dynamics, it should be noted that the definition of finite Blaschke
products restricts them to the unit disk. However, as previously mentioned, they extend
through the boundary as rational maps, holomorphic in all the Riemann sphere. The inter-
esting aspect about this is that this extension to the whole Riemann sphere is not arbitrary:
the behaviour of Blaschke products outside the unit disk is a mirror image of their behaviour
within it. This phenomenon is known as the reflection property (Proposition 1.4) and it stems
from an earlier result by Schwarz (Theorem A.4). Formally, it takes the form
B(z) =
1
B(1/z)
, for all z ∈ Cˆ.
Another important quality that can be hinted at first glance is that, for k = 1, finite Blaschke
products conform the family of automorphisms of D. Noteworthy, this can be generalised for
higher degree. In particular, for k ≥ 2, finite Blaschke products are precisely the family of holo-
morphic proper maps of the unit circle. In other words, they are the only holomorphic maps
Introduction vii
such that every point in D has exactly k preimages (counted with multiplicity). Another defin-
ing property of this family is their extension to the unit circle. Indeed, Fatou showed in 1923
(Theorem 1.6, [Fat23]) that any holomorphic self-map of D that can be extended continuously
to ∂D leaving both D and ∂D invariant is a finite Blaschke product.
Thus, it is clear that finite Blaschke products (of degree k ≥ 2) serve as an excellent common
thread in the exploration of complex dynamics. Now, the main question remains: what can we
say about the dynamics of this family?
Of course, the answer to this question depends in the characteristics of the Wolff-Denjoy
point, z0 ∈ D. According to this point, we can build a classification of finite Blaschke products
into four different types: elliptic, hyperbolic, simply parabolic and doubly parabolic. In the
elliptic case, z0 is inside the unit disk and orbits inside D converge to it, while those in Cˆ \D
tend to its reflected point 1/z0. In the other three cases, z0 ∈ ∂D, and hence all orbits in both
D and Cˆ \D converge to z0 = 1/z0. The difference between these three cases lies in nature of
this convergence.
With this classification, we are able to fully characterise the dynamics of finite Blaschke
products. In particular, we have the following result (Theorem 3.6).
(a)
z0
(b)
z0
(c)
z0
(d)
z0
1/z0
Figure 3: Characterisation of the dynamics of FBP. (a) shows the elliptic case, where the Fatou set has two
connected components, D and Cˆ \D. (b) and (c) show the hyperbolic and simply parabolic cases, respectively,
where the Fatou set has only one connected component and the Julia set is a Cantor set of ∂D. Finally, (d) shows
the doubly parabolic case, where the Fatou set has again two connected components, D and Cˆ \D.
Theorem (Characterisation of the dynamics of FBP). Let B be a finite Blaschke product of degree
d ≥ 2. Then, the following holds.
(a) If B is elliptic, then the Julia set of B is ∂D and the Fatou set of B consists of two invariant
connected components, D and Cˆ \D.
viii Introduction
(b) If B is hyperbolic, then the Julia set of B is a Cantor subset of ∂D and the Fatou set of B consists
of one invariant connected component.
(c) If B is simply parabolic, the Julia set of B is a Cantor subset of ∂D and the Fatou set of B consists
of one invariant connected component.
(d) If B is doubly parabolic, the Julia set of B is ∂D and the Fatou set of B consists of two invariant
Fatou components, D and Cˆ \D.
See Fig. 3. This concise yet conclusive result entirely determines the dynamical partition
of the Riemann sphere for Blaschke products. The Fatou set is composed of either one or two
invariant Fatou components, where all orbits converge to a fixed point. In contrast, the Julia
set is either connected, spanning the whole unit circle, or totally disconnected, being a Cantor
subset of ∂D. With this in mind, the natural next step towards characterising the dynamics of
this family of maps involves investigating their behaviour on the unit circle.
In this regard, previous work by Michael Shub with expanding maps from 1969 ([Shu69])
provides us with useful tools. Notably, Shub’s Theorem (Theorem 4.3) establishes a unique
semi-conjugacy between any finite Blaschke product of degree d, say B, and the map zd in the
unit circle. In other words, it guarantees the existence of a unique (not necessarily strictly)
monotonous, surjective and continuous map h : ∂D −→ ∂D such that h ◦ B(z) = (h(z))d for
z ∈ ∂D. Combining this result with the powerful properties of the Julia set, we conclude
that, in the cases where the Julia set spans the entire unit circle, this semi-conjugacy is in
fact a topological conjugacy, i.e. it is a continuous, bijective map with continuous inverse.
Specifically, we have the following (Theorem 4.4).
Theorem (Shub’s conjugacy). Let B be an elliptic or doubly parabolic finite Blaschke product of degree
d ≥ 2. Then, there exists a conjugacy to zd in ∂D.
However, the richness of this result extends further. Taking profit of the remarkable analytic
properties of finite Blaschke products, we can explicitly determine this conjugacy in the elliptic
and doubly parabolic case. Roughly speaking, we identify points whose orbit under B and zd
visit similar regions of the unit circle (see Fig. 4). This approach yields an explicit form of
Shub’s conjugacy through the itineraries of points, an instance of what is known as symbolic
dynamics.
At this point, one might wonder the relevance of this result within complex dynamics,
and whether we have ventured too far from dynamics into analysis. Far from it, this result
provides us with a conjugacy, the fundamental tool in the study of dynamics. As a matter
of fact, conjugacies preserve many essential dynamical properties, and hence, conjugating a
finite Blaschke product B to a simpler map such as zd is an excellent way of determining
the dynamics of B. For instance, periodic points are preserved through conjugacies, and it is
straight-forward to find that the periodic points of period n ≥ 1 of zd are precisely those of the
form
pn,k = e2πirn,k , where rn,k =
k
dn − 1, for k ∈ {1, . . . , d
n − 1} and k ̸ |dn − 1.
Introduction ix
B z4
p
1
z
z0
S(z) = S0(z0) = 144213 . . .
Figure 4: Sketch of the conjugacy between an elliptic or doubly parabolic FBP B of degree 4 and z4. We
divided the unit circle into four numbered regions, and identified points whose orbits visited the same regions.
The sequences S(z) and S0(z0) hold the information of which regions are visited.
Finally, once the behaviour of all orbits of finite Blaschke products is understood, one is
tempted to seek a generalisation of these results to a family of maps exhibiting a similar struc-
ture. One possibility is to consider functions that still adhere to the reflection property, but not
do not necessarily leave the unit disk invariant. This is accomplished by letting the zeros of
finite Blaschke products to be outside the unit disk. With this seemingly subtle modification,
most of the previous results no longer apply and the dynamics of Blaschke products undergoes
a significant transformation. The primary reason behind this drastic change lies in the exis-
tence of poles inside the unit disk, which invalidates the use of our main tool until now, the
Wolff-Denjoy Theorem. We refer to this new family of functions as generalised finite Blaschke
products, and they, despite the change, remain rational maps. Consequently, the Classification
Theorem still applies, affording us a glimpse into their emerging dynamics. See Fig. 5.
To delve into this exploration, we focus on a specific family of such generalised finite
Blaschke products, taking the form
Bθ,a = eiθz2
z− a
1− az , for θ ∈ R, a > 1.
In this project, using the power of the Classification Theorem and drawing on intriguing results
from number theory, we show that for certain pairs of parameters (θ, a), there exists an annular
region spanning around the unit circle on which the dynamics is essentially that of an irrational
rotation. This region is known as a Herman ring, and, even though they were first conjectured
to not exist for rational maps by Fatou and Julia themselves, their existence was proven in 1984
by Michael Herman in [Her84], using precisely generalised finite Blaschke products of odd
degree as an example.
Structure of the project
The basic results from complex analysis and topology are assumed throughout the work.
Nevertheless, we have included a short Appendix as a reminder, including some theorems and
definitions that appear recurrently throughout the project.
x Introduction
(a) (b)
Figure 5: Example of the Fatou sets of the Blaschke product Bw(z) = z3 z−w1−wz for w = a ∈ D (elliptic Blaschke
product, (a)), and for w = 1/a ∈ Cˆ \D (generalised Blaschke product, (b), for a = −0.1749 + 0.4129i. Iterates in
the green region converge to 0 and in the orange region to ∞, and the gradient represents the rate of convergence.
Note that the elliptic case is extremely simple, and the only common property between both cases is the reflection
property.
First, we devote Section 1 to the study of self-maps of the unit disk, and, in particular, to
finite Blaschke products. We include the most relevant properties of this family of maps, which
will be relevant when discussing their dynamics.
The second chapter is designed as a first introduction in complex dynamics. The Julia and
Fatou sets, fixed points and the different types of Fatou components are formally defined,
and the Classification Theorem is stated. We end the chapter with a review of the results
characterising iteration in the unit disk. Some, but not all, proofs are included, since general
theory is not the goal of the project.
In Chapter 3 we start discussing the dynamics of finite Blaschke products. We motivate
their classification and prove the theorem characterising their Julia and Fatou sets.
Then, Chapter 4 ties the loose end of the third chapter, the behaviour in the unit disk and
contains our main results. We introduce and prove Shub’s Theorem, and, with a short intro-
duction to symbolic dynamics, we explicitly find the conjugacy between elliptic and doubly
parabolic finite Blaschke products and zd in the unit circle.
Finally, the last chapter is to be taken as a short example that the dynamics of generalised
finite Blaschke products presents essential differences with respect to the ordinary ones. We
give some background in number theory and introduce a result determining the existence of
Herman rings for certain maps. Afterwards, we prove that, for certain parameters, a certain
family of degree 3 generalised finite Blaschke products satisfies the hypotheses of this result,
and hence has a Herman ring.
Chapter 1
Holomorphic self-maps of the unit disk
We develop this chapter to the study of holomorphic self-maps of the unit disk. Concerning
notation, we denote the unit disk as:
D = {z ∈ C : |z| < 1} ,
and refer to its boundary ∂D = {z ∈ C : |z| = 1} as the unit circle, and to its complement
Cˆ \D as the exterior disk.
First, we study the family of holomorphic automorphisms of D, which turns out to reduce
to two specific families of Möbius transformations. Subsequently, we introduce finite Blaschke
products as the products of these Möbius maps. Then, we study some of their analytic prop-
erties, such as zeros, fixed points and their derivatives. Finally, we consider a generalisation of
such maps, generalised finite Blaschke products.
For a deeper background in the concepts presented in this chapter, check [GMR18, Chapter
3], where most of the material in this chapter comes from.
1.1 Biholomorphisms of the unit disk
Our first step is to characterise holomorphic automorphisms of D, i.e. biholomorphisms.
We start by introducing two families of functions in D.
Definition 1.1 (Rotations and Blaschke factors). Let θ ∈ R and w ∈ D. We define the rotation
of angle θ and the Blaschke factor of zero w as the maps:
ρθ : D −→ D
z 7−→ eiθz
and βw : D −→ D
z 7−→ z− w
1− wz ,
respectively.
See Fig. 1.1 for a visual representation of these maps.
Theorem 1.2. Let AutH(D) be the set of biholomorphisms of D. Then,
AutH(D) = {ρθ ◦ βω : θ ∈ R,ω ∈ D} ,
where ρθ is the rotation of angle θ and βw is the Blaschke factor of zero w.
1
2 Holomorphic self-maps of the unit disk
(a)
(b)
(c)
ρθ
βw
Figure 1.1: Action of ρθ and βw over the unit disk. (a) represents the unit disk, (b) the image of the unit disk
under ρθ for θ = π/2 and (c) under βw for w = 3(1+ i)/10.
Proof. First, we shall see that maps of the form ρθ ◦ βw are biholomorphisms of D. It is clear
that for θ ∈ R, ρθ is a biholomorphism of the unit disk with inverse ρ−θ , since |ρθ(z)| =
∣∣eiθz∣∣ =
|z| < 1 and ρθ(ρ−θ(z)) = z for z ∈ D .
Next, for w ∈ D, note that βw is a Möbius transformation (see A.2). Hence, βw is biholo-
morphic in Cˆ. We have, for t ∈ R
∣∣∣βw(eit)∣∣∣ = ∣∣∣∣ eit − w1− weit
∣∣∣∣ = 1|eit|
∣∣eit − w∣∣
|e−it − w| = 1, (1.1)
and so, βw sends ∂D to ∂D. Since βw(w) = 0, we have that βw must send D to D. Thus, we
have that βw must be a biholomorphism of D. It is direct to see that its inverse is β−w.
It is left to see that any biholomorphism of D is of the form ρθ ◦ βw, for some θ ∈ R and
w ∈ D. Let f be a biholomorphism of D. In particular, it must have a holomorphic inverse
f−1. Let w = f−1(0) ∈ D and consider the Blaschke factor of zero −w i.e. β−w. Consider the
self-map of D defined as g = f ◦ β−w.
Clearly, g is well-defined, holomorphic and satisfies g(0) = f (β−w(0)) = f (w) = 0. Apply-
ing Schwarz Lemma A.1, we have |g(z)| ≤ |z| for all z ∈ D.
Moreover, since g is a composition of biholomorphisms, it must be, indeed, a biholo-
morphism. So, g−1 = βw ◦ f−1 is well-defined and holomorphic. Furthermore, it satisfies
g−1(0) = βw( f−1(0)) = βw(w) = 0. Thus, applying Schwarz Lemma A.1 again, we have∣∣g−1(z)∣∣ ≤ |z| for all z ∈ D. Hence, we must have, for all z ∈ D,
|z| =
∣∣∣g−1(g(z))∣∣∣ ≤ |g(z)| ≤ |z| , and so |g(z)| = |z| for all z ∈ D.
1.2 Proper self-maps of the unit disk 3
Thus, again as a consequence of Schwarz Lemma A.1, we have that for all z ∈ D g(z) = eiθz =
ρθ(z), for some θ ∈ R. Hence, we have f ◦ β−w = ρθ , and so f = ρθ ◦ βw, as we wanted to see.
1.2 Proper self-maps of the unit disk
Next, we move from bijective maps into higher degree self-maps of D.
Definition 1.3 (Finite Blaschke product). Let θ ∈ R and w1, . . . , wk ∈ D for some integer k ≥ 1.
The finite Blaschke product (FBP) of rotation θ and zeros w1, . . . , wk is the function B : D −→ D
defined as
B(z) = ρθ ◦
(
k
∏
l=1
βwl (z)
)
= eiθ
k
∏
l=1
z− wl
1− wlz . (1.2)
We call k ≥ 1 the degree of B.
It is clear from the last section 1.1 that B(D) = D and hence B is a holomorphic self-map
of D. In particular, finite Blaschke products of degree k = 1 are biholomorphisms of the form
ρθ ◦ βw.
Note that the definition of FBP restricts their domain to D. However, as these products are
indeed rational functions, they can be extended meromorphically to C, or, in fact, holomorphi-
cally to the Riemann sphere Cˆ. Furthermore, this extension is given as a reflection about the
unit circle. Formally, we have the following result.
Proposition 1.4 (Reflection property). Let B : D −→ D be a finite Blaschke product. Then, B can
be extended holomorphically to a rational function B : Cˆ −→ Cˆ, satisfying
B(z) =
1
B(1/z)
, ∀z ∈ Cˆ.
Proof. First, note that B is indeed a rational function (it is clear from Def. 1.3). It is left to see
that it satisfies the property of reflection. To do so, we shall find an appropriate map ϕ such
that ϕ ◦ B ◦ ϕ−1 satisfies the hypothesis of Schwarz Reflection Principle A.4.
Consider the Möbius transformation (see Appendix A.2):
ϕ(z) = −i z− 1
z + 1
, ϕ−1(z) =
i− z
i + z
Since Möbius transformation send circles to circles or straight lines, we have that ϕ(∂D) = R,
as ϕ(1), ϕ(i), ϕ(−i) ∈ R. Since ϕ(0) = i, we can conclude that ϕ(D) = H+ and ϕ(Cˆ \D) =
C \H+, where H+ :=
{
z ∈ Cˆ : Im(z) > 0} is the upper half plane (see Fig. 1.2).
Now, consider a FBP B : D −→ D. We can define f := ϕ ◦ B ◦ ϕ−1 : H+ −→ H+. As a
composition of holomorphic functions, f must be holomorphic in H+. Furthermore, since B
extends continuously to ∂D, f must do so for R. Then, by Schwarz Reflection Principle A.4, f
has a unique extension F : Cˆ −→ Cˆ defined as
F(z) = F(z), for z ∈ C \H+
4 Holomorphic self-maps of the unit disk
ϕ
H+D
ϕ−1
ϕ(0)
0
i
-1 1
ϕ(i) ϕ(1) ϕ(−1)
Figure 1.2: Sketch of the Möbius transformation ϕ : D −→H+.
Next, consider the function ϕ−1 ◦ F ◦ ϕ : Cˆ −→ Cˆ. Note that for z ∈ D, this function coincides
with B, in other words, it is an extension of B. As a consequence of the Identity Principle A.3,
it must be the only possible holomorphic extension. Furthermore, for z ∈ Cˆ \D, we have
ϕ−1 ◦ F ◦ ϕ(z) = ϕ−1 ◦ f ◦ ϕ(z) = ϕ−1 ◦ ϕ ◦ B ◦ ϕ−1 ◦ ϕ(z).
Finally, note that ϕ−1 ◦ ϕ(z) = 1/z, and so:
ϕ−1 ◦ F ◦ ϕ(z) = 1
B(1/z)
, for z ∈ Cˆ \D,
as we wanted to see.
Another interesting property of finite Blaschke products is properness, i.e. every point in
D has exactly k preimages, where k is the degree of the FBP. More details can be found in A.5.
Proposition 1.5. Let B be a finite Blaschke product of degree k. Then, B|D (resp. B|∂D and B|Cˆ\D) is a
proper self-map of degree k of D (resp. of ∂D and Cˆ \D).
Proof. We want to see that every point in D, ∂D or Cˆ \D has exactly k preimages in the same
set, counted with multiplicity.
Hence, consider a ∈ C. The preimages of a are exactly the solutions of the equation B(z) =
a. Writing B in the form (1.2) for some θ ∈ R, w1, . . . , wk ∈ D, the equation B(z) = a is
equivalent to
eiθ
k
∏
l=1
(z− wl) = a
k
∏
l=1
(1− wlz),
which is a polynomial equation of degree k. Then, a must have exactly k preimages in C,
counted with multiplicity.
Next, we have to see that these preimages are in the same set as a. Recall that B(D) = D.
Hence, applying the Reflection Property 1.4, we know that:
B(z) =
1
B(1/z)
∈ Cˆ \D, for z ∈ Cˆ \D,
1.2 Proper self-maps of the unit disk 5
B(z) =
1
B(z)
and so |B(z)|2 = 1, for z = 1
z
∈ ∂D.
In other words, B(D) = D, B(∂D) = ∂D and B(Cˆ \D) = Cˆ \D. Then, if a ∈ D (or ∂D, Cˆ \D),
all k preimages of a must also be in D (or, respectively, ∂D, Cˆ \D), as we wanted to see.
For the point a = ∞ ∈ Cˆ, note that their preimages are exactly the poles of B, and these are
the k values 1/wl ∈ Cˆ \D.
Therefore, finite Blaschke products of degree k ≥ 2 are surjective self-maps of the unit disk
D, but they are not one-to-one. In fact, they cover D exactly k times. We will now see that, in
the same way as FBP of degree 1 generated the biholomorphisms of D, FBP are precisely the
family of proper self-maps of D.
Theorem 1.6 (Fatou’s Theorem). Let f : D −→ D be a holomorphic self-map of the disk into itself
such that
lim
|z|→1−
| f (z)| = 1
Then, f is a finite Blaschke product. Consequently, any holomorphic proper self-map of D is a FBP.
Proof. As a consequence of the Identity Theorem A.3, if f has infinitely many zeros in D, they
must accumulate in ∂D. However, as lim|z|→1− | f (z)| = 1 ̸= 0, f must have a finite number of
zeros in D. Let us consider a finite Blaschke product B with the same zeros as f , counted with
multiplicity, of arbitrary rotation. Then, we can consider the map h : D −→ C defined as
h(z) =
limw→z
f (w)
B(w) , if B(z) = 0
f (z)
B(z) , otherwise.
Since f and g have the exact same zeros, the function h is well-defined, holomorphic and differ-
ent from zero in all D. This way, the function 1/h is also holomorphic in all D. Furthermore,
lim
|z|→1−
|h(z)| = lim
|z|→1−
| f (z)|
|B(z)| = 1, and lim|z|→1−
1
|h(z)| = lim|z|→1−
|B(z)|
| f (z)| = 1.
So, by the Maximum Modulus Principle A.2, both |h(z)| ≤ 1 and |1/h(z)| ≤ 1 for all z ∈ D.
This means that | f (z)| / |B(z)| = |h(z)| = 1, and so, f /B must be a constant function of
modulus 1, i.e., f (z)/B(z) = eiθ ∈ ∂D for some θ ∈ R. In conclusion, f (z) = eiθB(z) for all
z ∈ D and so f is a finite Blaschke product.
Finally, consider an arbitrary holomorphic proper map f : D −→ D of degree k and let
(zn)n be a sequence tending to ∂D, in other words, |zn| −→ 1.
Now, if the image sequence ( f (zn))n ⊆ D does not accumulate in ∂D, then it must be
contained in a compact set K ⊆ D. In this case, since f is proper, the set f−1(K) ⊆ D must also
be compact. However, this would mean that the original sequence (zn)n ⊆ f−1(K) must have a
convergent sub-sequence in f−1(K) ⊆ D, contradicting the original hypothesis. In conclusion,
( f (zn))n must accumulate in ∂D, and so, | f (zn)| −→ 1.
In other words, the limit of | f (z)| as |z| −→ 1 is 1. Hence, by Fatou’s Theorem 1.6, f must
be a finite Blaschke product.
6 Holomorphic self-maps of the unit disk
In conclusion, we just saw that any map defined in D that maps sequences converging
to ∂D to sequences converging to ∂D must have an analytic continuation to D (and in fact,
holomorphic in Cˆ) and has to be a finite Blaschke product.
1.3 Derivatives of finite Blaschke products in the unit circle
Since one of our goals is to study the behaviour of finite Blaschke products on the unit circle
∂D, it is useful to study the derivative of a general FBP. For this regard, when working with
large products of functions in the Riemann Sphere f = f1 · · · fk, one easier way to compute
derivatives is to look at the fraction f ′/ f , as we have
f ′ =
k
∑
l=1
f1 · · · fl−1 f ′l fl+1 · · · fk =
k
∑
l=1
f ′l
f
fl
=⇒ f
′
f
=
k
∑
l=1
f ′l
fl
.
This quotient is known as the logarithmic derivative of f , and for a FBP of the form in Eq.1.2,
it takes the form
B′(z)
B(z)
=
k
∑
l=1
(
z− wl
1− wlz
)′ 1− wlz
z− wl =
k
∑
l=1
1− |wl |2
(1− wlz)(z− wl) , (1.3)
for all z ∈ Cˆ, where we used that the derivative of a general Möbius transformation M(z) =
az+b
cz+d is M
′(z) = ad−bc
(cz+d)2 , for ad− bc ̸= 0. This has interesting implications.
Proposition 1.7. Let B be a finite Blaschke product. Then, for all z ∈ Cˆ
B′(z) =
B(z)2
z2
B′(1/z)
Proof. We write B in the form described in Eq. (1.2) and then evaluate Eq. (1.3) for 1/z. We
have
B′(1/z)
B(1/z)
=
k
∑
l=1
1− |wl |2
(1− wl/z)(1/z− wl) = z
2
k
∑
l=1
1− |wl |2
(z− wl)(1− wlz) = z
2 B′(z)
B(z)
.
Changing z for 1/z and using the Reflection Property 1.4, we get the expression we were
looking for.
Proposition 1.8 (FBP in the unit circle). Let B be a finite Blaschke product of degree k ≥ 1 and θ ∈ R.
Then,
(a) B′(eiθ) ̸= 0. Consequently, every point in ∂D has exactly k distinct preimages in ∂D.
(b) If B(eiθ) = eiθ , we have B′(eiθ) ∈ R>0.
(c) If arg is a branch cut of the argument of the curve B(eit) for t ∈ R, we have ddt arg B(eit) =∣∣B′(eit)∣∣ > 0.
(d) B is a local diffeomorphism of ∂D.
1.4 Generalisations of finite Blaschke products 7
Proof. (a) Since B(∂D) = ∂D, we have that
∣∣B(eiθ)∣∣ = 1 for all θ ∈ R. With this, having B as
in Eq. (1.2), we can use Eq. (1.3), and thus∣∣∣B′(eiθ)∣∣∣ = ∣∣∣∣B′(eiθ)B(eiθ)
∣∣∣∣ =
∣∣∣∣∣ k∑l=1 1− |wl |
2
(1− wleiθ)(eiθ − wl)
∣∣∣∣∣ =
=
∣∣∣∣∣ k∑l=1 1eiθ 1− |wl |
2
(e−iθ − wl)(eiθ − wl)
∣∣∣∣∣ =
∣∣∣∣∣ 1eiθ k∑l=1 1− |wl |
2
|eiθ − wl |2
∣∣∣∣∣ =
=
k
∑
l=1
1− |wl |2
|eiθ − wl |2
> 0.
since wl ∈ D for all l.
Next, let a ∈ ∂D. Due to Proposition 1.5, we know that a has k preimages in ∂D. It is
left to see that these are distinct, or, equivalently, that their multiplicity as solutions of
the equation B(z) = a is exactly 1. However, if one of its preimages a∗ had multiplicity
greater than one, we would have B′(a∗) = 0, which cannot happen since a∗ ∈ ∂D.
(b) Let θ ∈ R such that B(eiθ) = eiθ . Assuming B is in the form (1.2), by Eq. (1.3), we have
B′(eiθ) = eiθ
k
∑
l=1
1− |wl |2
eiθ(e−iθ − wl)(eiθ − wl) =
k
∑
l=1
1− |wl |2
|eiθ − wl |2
∈ R+.
(c) First, note that since B(∂D) = ∂D, the curve B(eit) does not cross 0, and so there exists a
differentiable branch cut of the argument of this curve for t ∈ R. In other words, we can
write B(eit) = eiψ(t) for some differentiable function ψ : R −→ R. Hence, arg B(eit) = ψ(t)
is well-defined and differentiable.
Now, we can assume B is in the form (1.2) and use expression 1.3. We have, then,
ψ′(t) = eit
B′(eit)
B(eit)
= eit
k
∑
l=1
1− |wl |2
(1− wleit)(eit − wl) =
k
∑
l=1
1− |wl |2
|eit − wl |2
=
∣∣∣B(eit)∣∣∣ ,
which must be positive as we saw in (a).
(d) This result is a simple corollary of the Inverse Function Theorem, since, by (a), B′(z) ̸= 0
for every z ∈ ∂D.
In fact, one can see that a FBP B of degree k is what is known as a degree k covering of ∂D,
meaning that ∂D is divided into k arc sections, each one being sent diffeomorphically by B to
the whole ∂D. More details will be given in Section 4.2.2.
1.4 Generalisations of finite Blaschke products
Finally, we end this chapter by considering a simple generalisation of finite Blaschke prod-
ucts by allowing their zeros to be outside of the unit disk. In particular, we can define the
following.
8 Holomorphic self-maps of the unit disk
B
z1
z2
z3
z4
a
Ia
Iz1
∂D ∂D
Figure 1.3: Proposition 1.8(d). Every finite Blaschke product B is diffeomorphic between neighbourhoods of any
a ∈ ∂D and one of its preimages zj ∈ ∂D.
Definition 1.9 (Generalised finite Blaschke product). Let θ ∈ R and w1, . . . , wk ∈ Cˆ satisfying
wlwr ̸= 1 for l, r ∈ {1, . . . , k}.
We say that the generalised finite Blaschke product (GFBP) of rotation θ and zeros w1, . . . , wk
is the function B : Cˆ −→ Cˆ defined as
B(z) = eiθ
k
∏
l=1
z− wl
1− wlz . (1.4)
Remark 1.10. Note that, if w ∈ Cˆ \D, we have
z− w
1− wz =
w
w
(1/w)z− 1
(1/w)− z = γ
1− az
z− a , for γ =
w
w
∈ ∂D and a = 1
w
∈ D
Then, if two zeros of a generalised finite Blaschke product satisfy wlwr = 1, for l ̸= r, we have
that the factors corresponding to wl and wr cancel out, leaving only a constant rotation. On the
other hand, if l = r, we have that wlwl = |wl |2 = 1, and so, wl = eiθ and
z− eiθ
1− e−iθz = −e
iθ 1− e−iθz
1− e−iθz = −e
iθ ∈ ∂D.
In other words, if any wl ∈ ∂D, then its corresponding factor is a constant rotation. Then, the
condition wlwr ̸= 1 guarantees that no factors cancel out and so a GFBP of the form Eq.(1.4) is
a rational function of degree k.
We just saw that every factor of the product with a zero in Cˆ \D is the inverse of a Blaschke
factor with a zero in D. So, one can think of GFBP as the quocient of two finite Blaschke
products with no common zeros.
This implies, in particular, that the Reflection Property 1.4 is still satisfied for GFBP, and
they still send ∂D onto ∂D. However, most other properties of finite Blaschke products are
not inherited for the generalised ones, since they are a consequence of Schwarz Lemma A.1,
which cannot be applied now as GFBP do not sendD intoD. In fact, whenever we have a zero
outside of D, we have a pole in D, and so there is a neighbourhood of this point being sent to
a neighbourhood of infinity.
Nevertheless, we can still find a useful equivalence for these maps. We have the following
result.
1.4 Generalisations of finite Blaschke products 9
Proposition 1.11. Let R : Cˆ −→ Cˆ be a rational map such that R(∂D) ⊆ ∂D. Then, R is a generalised
Blaschke product.
Proof. The proof of this result is analogous to the one of Fatou’s Theorem 1.6, considering now
a generalised FBP with the same zeros and poles inside of D of B. This way, the quotients R/B
and B/R are holomorphic in D and the same argument can be used.
Chapter 2
Complex dynamics
In broad terms, dynamical systems encompass the study of functions dependent on a tem-
poral parameter. The most natural example, responsible for the nomenclature in this field, is
the system of trajectories arising in the study of autonomous differential equations, which have
been known to govern the movements of planets since the seventeenth century.
Three hundred years later, G. Julia and P. Fatou kick-started the study of complex dynamical
systems. Here, the object of study shifts from differential equations to the iteration of holomor-
phic maps, and the continuous temporal parameter changes to a discrete index, labelling the
iterates. Now, instead of stellar orbits, the robust properties of holomorphic functions create
complex, intriguing structures with remarkable mathematical properties.
This chapter aims to formally describe these constructions and provide a preliminary un-
derstanding of these complex dynamical systems. The contents of this chapter can be found in
[CG93, Chapters 2, 3], [Ste11, Chapters 2, 3] or [Mil90, Sections 3 to 14].
Concerning notation, given a rational map R(z), we denote the n-th iterate as Rn(z) =
R(Rn−1(z)), where n ≥ 1 and we are denoting R0(z) = z. This should not be confused with
the n-th derivative of R, which is denoted as R(n)(z).
2.1 The dynamic partition of the Riemann Sphere: Fatou and Julia
sets
The fundamental elements of complex dynamics are the Julia and Fatou sets, which bear the
names of the early developers of this field. These divide the Riemann Sphere into two regions
with different dynamical properties, and they are the natural starting point in the study of
holomorphic iteration.
For our purpose, we can limit ourselves to the case of iteration of holomorphic maps in
the whole Riemann Sphere, Cˆ, which are precisely rational maps (see Appendix A.2). All this
results are discussed in [Ste11, Chapter 2].
Before starting with iteration, let us recall the concept of degree of a rational map. Given
a rational map R(z) = P(z)/Q(z), for P and Q relatively prime polynomials, the degree of R
is the maximum between deg P and deg Q. As discussed in A.5, it is straight forward to see
that this is the topological degree of R, i.e., that every point in Cˆ has exactly d preimages in
10
2.1 The dynamic partition of the Riemann Sphere: Fatou and Julia sets 11
Cˆ, counted with multiplicity. In other words, R is a proper map of degree d. Notice that in
the case of degree d = 0, R is a constant map; and in the case of degree d = 1, R is a Möbius
transformation. Hence, we focus on rational maps of degree d ≥ 2.
Let us start discussing rational iteration by introducing a simple but relevant example, the
quadratic family, which comprises rational maps of degree d = 2 of the form Qc(z) = z2 + c
for some c ∈ C.
Example 2.1. Let us consider the rational function Q0(z) = z2. If we study its family of iterates,
we can see that Qn0(z) = Q0 ◦
(n)· · · ◦ Q0(z) = z2n . By writing z in polar coordinates z = reiθ , we
can see that
Qn0(z) = r
2n ei2
nθ n−→
{
∞, if r > 1
0, if r < 1,
(2.1)
which induces a division of the complex plane into open regions where the iterates converge
to certain points, separated by a closed set.
This division in invariant open sets with the same dynamics is in fact a general trait of
holomorphic maps.
Definition 2.2 (Invariant set). Let R be a rational map. We say a set U ⊆ Cˆ is forward invariant
if R(U) ⊆ U, backward invariant if R−1(U) ⊆ U and completely invariant if U is forward and
backward invariant.
Definition 2.3 (Normality). Given a rational map R of degree d ≥ 2 and let U ⊆ Cˆ be an open,
forward invariant set. We say that the family of iterates of R is normal in U if any partial
sequence of {Rn}n has a sub-sequence which converges uniformly to a holomorphic map in
compact subsets of U. Given z ∈ Cˆ, we say the family of iterates of R is normal in z if it is
normal in some neighbourhood of z.
Remark 2.4. Note that since we are considering rational maps, U ⊆ Cˆ and the iterates can
converge to ∞.
One result characterising normal families with historical relevance is Montel’s theorem,
which served as the cornerstone for the development of the theory of rational iteration.
Theorem 2.5 (Montel’s Theorem). Let R be a rational map. Let U ⊆ R be an open set. If Cˆ \⋃
n≥0 Rn(U) contains at least 3 points, the family of iterates of R is normal in U.
Definition 2.6 (Fatou and Julia sets). Let R be a rational map of degree d ≥ 2. We define the
Fatou set of R as
F (R) = {z ∈ Cˆ : the iterates of R are normal at z} ,
and the Julia set of R as its complement:
J (R) = {z ∈ Cˆ : the iterates of R are not normal at z} .
12 Complex dynamics
Note that in general, we omit the dependence on R and denote J = J (R),F = F (R).
Example 2.7. In our previous example, we saw that {Qn0}n converges to the constant (holomor-
phic) maps 0 in D and to ∞ in ∂D. Hence, the iterates of Q0 are normal in D and Cˆ \D.
However, iterates are not normal in any open set U containing points of ∂D, since this set
would contain points from D, with iterates converging to 0, and from Cˆ \D, with iterates con-
verging to ∞. Hence, not any sub-sequence of the iterates of Q0 can converge to a holomorphic
map in U.
In conclusion, J (Q0) = ∂D and F (Q0) = Cˆ \ ∂D.
Hence, the map Q0 exemplifies this partition of Cˆ. Now, we shall introduce some basic
properties of these sets.
Proposition 2.8 (Properties of Julia and Fatou sets). Let R be a rational map of degree d ≥ 2. Let
J and F be its Julia and Fatou sets, respectively. Then,
(1) F and J are completely invariant.
(2) F is open and J is closed.
(3) For all k ≥ 1, F (Rk) = F (R) and J (Rk) = J (R).
(4) J is not empty.
Proof. (1) As F and J form a partition of the Riemann Sphere, it is enough to see that
R−1(F ) ⊆ F and that R−1(J ) ⊆ J .
For the first inclusion, let p ∈ R−1(F ), i.e., R(p) ∈ F . Then, there exists an open neigh-
bourhood of R(p), U, such that the family of iterates of R is normal in U. Now, since R is
continuous, R−1(U) is an open neighbourhood of p. Furthermore, the family of iterates
of R in R−1(U) coincides exactly with the family of iterates of R in U. Hence, the iterates
are normal in R−1(U). In other words, R−1(F ) ⊆ F .
Second, let p ∈ R−1(J ). Then, for every open neighbourhood U of p, R(U) is an open
neighbourhood of R(p) ∈ J , and so, the family of iterates of R is not normal in R(U)
and thus cannot be normal in U. This means R−1(J ) ⊆ J .
(2) Let z ∈ F . By definition, it exists an open neighbourhood of z, U, where the family {Rn}n
is normal. Note that as U is open, U is a neighbourhood of all its points, and so, all their
points are normal, i.e. U ⊆ F , and F is open.
As J = Cˆ \ F , we have that J is closed.
(3) We shall study the normality of the families of iterates of R, {Rn}n, and of Rk,
{
Rnk
}
n.
First, note that
{
Rnk
}
n ⊆ {Rn}n. Hence, if {Rn}n is normal in a domain U,
{
Rnk
}
n must
be so. For the converse, note that for all k ≥ 1, we can divide the family of iterates in the
following way:
{Rn}n≥0 =
{
Rnk
}
n≥0
∪
{
R1 ◦ Rnk
}
n≥0
∪ · · · ∪
{
Rk−1 ◦ Rnk
}
n≥0
.
2.1 The dynamic partition of the Riemann Sphere: Fatou and Julia sets 13
This way, if we assume the family of iterates of Rk,
{
Rnk
}
n, is normal in a domain U and
consider any sub-sequence
{
Rnlk
}
l converging uniformly to a holomorphic map f in U,
then, the sequence
{
Rj ◦ Rnlk}l must converge uniformly to Rj ◦ f , since Rj is uniformly
continuous for all j ∈ {0, . . . , k− 1} (Rj is a continuous map defined in a compact set, the
Riemann sphere). Hence, {Rn}n is normal in U if and only if
{
Rnk
}
is normal in U. This
means that the Fatou sets of R and Rn are equal, and thus, so are the Julia sets.
(4) Assume J is empty. Then, R is normal in Cˆ, and so, a sub-sequence of the iterate family,
{Rnk}k, must converge uniformly to a holomorphic map f : Cˆ −→ Cˆ. Hence, f must a
rational map of degree d0 ≥ 0.
First, assume d0 = 0 and f (z) = c is a constant map, for some c ∈ Cˆ. Then, since {Rnk}k
converges uniformly to c, there exists k0 ≥ 0 such that the image of Rnk is contained in
a neighbourhood of c for all k ≥ k0. However, this cannot happen since Rn is a rational
map, and so it is onto Cˆ for all n ≥ 0.
Second, assume d0 ≥ 1. Then, f has d0 zeros or d0 poles. Consider now any differentiable
curve γ : (0, 1) −→ Cˆ avoiding a neighbourhood of the zeros and poles of f . Then, there
exists ε = inft f (γ(t)) > 0.
Now, as {Rnk}k tends to f uniformly, there must exist k0 ≥ 0 such that |Rnk(z)− f (z)| <
ε ≤ | f (z)| for all k ≥ k0. Then, by Rouché’s Theorem A.5, the index of Rnk must be equal
to the index of f for k ≥ k0. Since γ is arbitrary, by the Argument Principle A.6, Rnk must
have the same number of zeros and poles as f , and so, the same degree. However, the
degree of Rnk is equal to nkd for each k, contradicting the hypotheses that f has a finite
degree.
In conclusion, J cannot be empty.
With these general properties, we just saw that the Julia and Fatou sets indeed induce a
dynamical partition of the Riemann Sphere into two completely invariant sets. Now, we give
more properties further characterising this sets. Since their proofs follow a similar structure as
those of Proposition 2.8 but involve wider context, we do not include them. However, they can
be found in [Ste11, Theorem 2.4.3], [Ste11, Theorem 2.4.4] and [Ste11, Theorem 2.5.2].
Proposition 2.9 (Properties of Julia and Fatou sets, II). Let R be a rational map of degree d ≥ 2 and
J its Julia set. Then:
(5) For every point a ∈ J , J = ⋃n≥0 R−n(a)
(6) J is perfect, i.e., it has no isolated points
(7) J has infinitely many elements
(8) (Blow-up property) For every domain D intersecting the Julia set, there exists a n ≥ 0 such that
Rn(D ∩ J ) = J
14 Complex dynamics
J
J ∩ D
Rn(J ∩ D)
D(a) (b)
Figure 2.1: Dynamical plane of the map Qi(z) = z2 + i. The red region indicates convergence to ∞ (a). The Julia
set is isolated in panel (b), and an arbitrary domain intersecting the Julia set is shown in a red square. For a certain
n ≥ 0, the image of the Julia set inside this domain generates again the whole Julia set.
2.2 Local theory
The Julia and Fatou sets we just introduced will be the basic framework for studying the
dynamics of rational maps. To fully understand their characteristics, it is necessary to study
the behaviour of the iterates in some sets. Our main tool are conjugacies.
Definition 2.10 (Conjugacies). Let R, R′ be rational maps. Let U, V ⊆ Cˆ be domains. We
say that R and R′ are (conformally) conjugate in U or V if there exists a biholomorphic map
h : U −→ V such that R′ = h ◦ R ◦ h−1 in V. We say h is a (conformal) conjugacy.
If h is not biholomorphic but it is a homeomorphism, we say R and R′ are topologically
conjugate and that h is a topological conjugacy.
Our aim is to conjugate the dynamics of rational maps to simpler functions on a neighbour-
hood of certain points. We follow the classification of [CG93, Chapter II].
Definition 2.11 (Fixed point). Let R be a rational map. We say z0 ∈ Cˆ is a fixed point of R if
R(z0) = z0. In this case, we define the multiplier of z0 as λ = R′(z0).
Definition 2.12 (Periodic point). Let R be a rational map. We say z0 ∈ Cˆ is a periodic point
of period p of R if z0 is a fixed point of Rp for minimal p ∈ Z≥0. In this case, we say that
α =
{
z0, R(z0), . . . , Rp−1(z0)
}
is a periodic cycle of period p, and its multiplier is λ = (Rp)′(zj)
for any j ∈ {0, . . . , p− 1}.
Note that every element in any periodic cycle is in fact a periodic point of the same period.
Furthermore, the multiplier is well-defined since, by the Chain Rule,
λ = (Rp)′(z0) =
p−1
∏
j=0
R′(Rj(z0)) =∏
z∈α
R′(z).
2.2 Local theory 15
Multipliers determine the local behaviour of fixed points and periodic cycles, and allow us to
establish a classification amongst them.
Definition 2.13 (Classification of cycles). Let R be a rational map and α a periodic cycle or fixed
point of R with multiplier λ. We say α is
(i) super-attracting if λ = 0,
(ii) attracting if 0 < |λ| < 1,
(iii) indifferent if |λ| = 1 (if λ = e2πir with r ∈ Q, we say α is parabolic),
(iv) repelling if |λ| > 1.
Notice that, due to Property (3) in Proposition 2.8, J (Rp) = J (R) and F (Rp) = F (R) for
all p ≥ 1. Since periodic points of period p of R are fixed points of Rp, we can reduce to the
study of the dynamics of rational maps around fixed points.
2.2.1 Attracting and repelling case
We shall treat the attracting and repelling case together, as they are two faces of the same
coin. In fact, let R be a rational map and z0 ∈ Cˆ a repelling fixed point with multiplier
R′(z0) = λ with |λ| > 1. Since λ ̸= 0, there exists a neighbourhood of z0 where R has
holomorphic rational inverse R−1. Hence, z0 is now an attracting fixed point of R−1 with
multiplier 1/λ.
The result characterising these fixed points is the Koenigs-Schröder Theorem.
Theorem 2.14 (Koenigs-Schröder). Let R be a rational map of degree d ≥ 2 and z0 ∈ Cˆ an attracting
or repelling fixed point of R. Then, R is locally conjugate to λz, where λ ∈ Cˆ.
In other words, there exist U neighbourhood of z0, V neighbourhood of 0, and a biholomorphism
ϕ : U −→ V such that
ϕ ◦ R(z) = λϕ(z), ∀z ∈ U.
Moreover, ϕ is unique up to multiplicative constant.
Proof. This proof follows the outline of [Ste11, Theorem 3.4.1]. First, we consider the attracting
case. Let us start by proving uniqueness. Assume ϕ : U −→ V is a diffeomorphism satisfying
ϕ ◦ R = λϕ, defined in some appropriate neighbourhoods U,V of z0 and 0, respectively. If we
evaluate the conjugacy expression on z0, we get ϕ(z0) = λϕ(z0), and so ϕ(z0) = 0. Furthermore,
since ϕ is a diffeomorphism, we have ϕ′(z0) ̸= 0. Hence, we can divide ϕ by the constant ϕ′(z0)
and get an equivalent diffeomorphism ψ with ψ′(z0) = 1.
Assume then, without loss of generality, ϕ′(z0) = 1 and ϕ(z0) = 0. We have
ϕ ◦ Rn(z) = λnϕ(z) −→ 0 for z ∈ U, as |λ| < 1.
Then, as ϕ is a diffeomorphism, Rn(z) −→ ϕ−1(0) = z0 for z ∈ U. This way, we can write
ϕ′(0) = lim
z→z0
ϕ(z)− ϕ(z0)
z− z0 = limn→∞
ϕ(Rn(z))− 0
Rn(z)− z0 = 1, by hypothesis.
16 Complex dynamics
And so, by isolating ϕ(z) from the conjugacy equation,
ϕ(z) =
ϕ(Rn(z))
λn
∀n ⇒ ϕ(z) = lim
n→∞
ϕ(Rn(z))
λn
= lim
n→∞
Rn(z)
λn
.
Which means that ϕ is uniquely determined. This proof also gives us intuition on how to
prove the existence of this conjugacy. In fact, we will prove that this limit exists and satisfies
our hypotheses.
Obs. 1. There exists a forward invariant neighbourhood of z0, U.
This can be seen by considering the definition of derivative. Indeed, since R′(z0) = λ, for
every ε > 0, there exists a δ > 0 such that∣∣∣∣∣∣∣∣R(z)− R(z0)z− z0
∣∣∣∣− |λ|∣∣∣∣ < ε, for z ∈ D(z0, δ).
Now, if we choose ρ such that |λ| < ρ < 1, we can set ε = ρ − |λ| > 0, and so, for every
z ∈ D(z0, δ) ∣∣∣∣R(z)− R(z0)z− z0
∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣R(z)− R(z0)z− z0
∣∣∣∣− |λ|∣∣∣∣+ |λ| < ρ− |λ|+ |λ| = ρ.
Hence, for z ∈ D(z0, δ), |R(z)− R(z0)| < ρ |z− z0| < |z− z0| < δ, i.e., R(z) ∈ D(R(z0), δ) =
D(z0, δ), since z0 is a fixed point of R. Then, for any ρ in (|λ| , 1), there exists a δ such that
U = D(z0, δ) is forward invariant, as we wanted to see.
Now, assume a fixed ρ and its corresponding neighbourhood of z0, U = D(z0, δ) for some
δ > 0. In this set, we can define
ϕn(z) :=
Rn(z)− z0
λn
, for z ∈ U.
Obviously, these maps are holomorphic and send U to a neighbourhood of 0, Vn = ϕn(U). Our
aim is to show that these maps comprise a Cauchy sequence. For this purpose, consider
|ϕn+1(z)− ϕn(z)| =
∣∣∣∣R(Rn(z))− z0λn+1 − Rn(z)− z0λn
∣∣∣∣ = |R(w)− z0 − λ(w− z0)||λn+1| .
for w = Rn(z).
Obs. 2. There exists C < 0 such that |R(z)− z0 − λ(z− z0)| < C |z− z0|2.
This is direct if we consider the Taylor series of R(z),
R(z) = z0 + λ(z− z0) + o(z2)⇒ |R(z)− z0 − λ(z− z0)| < C |z− z0|2 .
for some C > 0.
Obs. 3. For each n ≥ 1, |Rn(z)− z0| < (|λ|+ Cδ)n |z− z0|.
This is a direct consequence of Obs. 2. We have
|R(z)− z0| ≤ |R(z)− z0 − λ(z− z0)|+ |λ(z− z0)| < (Cδ+ |λ|) |z− z0| .
Now, if it is true for n ≥ 1, then,∣∣∣Rn+1(z)− z0∣∣∣ = |R(Rn(z))− z0| < (|λ|+ Cδ) |Rn(z)− z0| < (|λ|+ Cδ)n+1 |z− z0| .
2.2 Local theory 17
With these two observations, we get
|ϕn+1(z)− ϕn(z)| < C|λ|n |R
n(z)− z0|2 < C|λ|
(
(|λ|+ Cδ)2
|λ|
)n
|z− z0|2 .
Next, as we can choose δ as small as we want, and |λ|2 < |λ|, we can choose δ < 0 such that
ξ :=
(|λ|+ Cδ)2
|λ| < 1.
Hence, we have that |ϕn+1(z)− ϕn(z)| < (C/ |λ|)ξn |z− z0|2. Finally, this means that for m ≥
n ≥ 0,
|ϕm(z)− ϕn(z)| < C|λ| (ξ
n + ξn+1 + · · ·+ ξm−1) |z− z0|2 .
which must tend to 0 uniformly for all z ∈ U since the series ∑n≥1 ξn is convergent. Then,
the sequence of maps ϕn must converge uniformly to a limit ϕ : U −→ ϕ(U), which must be
holomorphic since all ϕn are. Furthermore, since for all n ≥ 1, ϕ′(z0) = (R′(z0))n/λn = 1, we
have that ϕ is a biholomorphism in a neighbourhood of z0 inside U.
Lastly, if z0 is repelling, then R−1 is defined in some neighbourhood of z0 and has z0 as an
attracting fixed point. So, the previous arguments can be applied.
Therefore, around each attracting point there exists a neighbourhood in which points con-
verge to it under iteration. In particular, attracting points are in the Fatou set, and we can
define the following.
Definition 2.15 (Attracting basin). Let R be a rational map of degree d ≥ 2 and let z0 ∈ Cˆ be
an attracting fixed point of R. We define the basin of attraction of z0, A(z0) as the set of points
whose orbit converges to z0, i.e.
A(z0) =
{
z ∈ Cˆ : Rn(z) −→ z0
}
.
2.2.2 Super-attracting case
The case of super-attracting fixed points is similar to the attracting case.
Theorem 2.16. Let R be a rational map of degree d ≥ 2 and z0 ∈ Cˆ a super-attracting fixed point
of R. Let k ≥ 0 be the minimal such that R(k)(z0) ̸= 0. Then, R is locally conjugate to azk, where
a = R(k)(z0) ̸= 0.
In other words, there exist U, V ⊆ Cˆ neighbourhoods of z0 and 0, respectively, and a biholomorphism
ϕ : U −→ V such that
ϕ ◦ R(z) = aϕ(z)k, ∀z ∈ U.
Moreover, ϕ is unique up to multiplicative constant.
The proof of this result can be found in [Ste11, Theorem 3.3.1]. It is similar in spirit to the
Koenigs-Schröder Theorem 2.14 and we do not include it here.
Similarly to the attracting case, we have that super-attracting points have a neighbourhood
where the iterates are normal.
18 Complex dynamics
Definition 2.17 (Super-attracting basin). Let R be a rational map of degree d ≥ 2 and let z0 ∈ Cˆ
be a super-attracting fixed point of R. We define the basin of attraction of z0, A(z0) as the set
of points iterating to z0, i.e.
A(z0) =
{
z ∈ Cˆ : Rn(z) −→ z0
}
.
Hence, super-attracting points also belong to the Fatou set.
2.2.3 Parabolic case
Indifferent fixed points have more subtleties to take into account, discussed widely in
[Bea91, Section 6.5] and [Ste11, Section 3.5]. In fact, here we only consider parabolic points
for which the multiplier of the fixed point is exactly 1. In other words, the rational map R has
expansion near z0 as
R(z) = z0 + (z− z0) + o(|z− z0|2) = z + o(|z− z0|2).
Before characterising the dynamic in a neighbourhood of z0, we need to define some additional
concepts.
Definition 2.18 (Multiplicity). Let R be rational map of degree d ≥ 2 and let z0 ∈ Cˆ be a
parabolic fixed point. We say the multiplicity of z0 is the minimal n ≥ 2 such that R(n)(z0) ̸= 0,
where R(n) is the n-th derivative of R.
Remark 2.19. Normally, we denote the multiplicity of a fixed point as n+ 1, to further indicate
that R(k)(z0) = 0 for all k with 2 ≤ k ≤ n.
Definition 2.20 (attracting/repelling vectors). Let R be a rational map of degree d ≥ 2 and let
z0 ∈ Cˆ be a parabolic fixed point of multiplicity n + 1, with a = R(n+1)(z0) ̸= 0.
We define the attracting vectors of R as the n different n-th roots of −1/(an), and the
repelling vectors of R as the n-th roots of 1/(an).
Note that as R′(z0) = 1 ̸= 0, there is a neighbourhood of z0 where R−1 is well-defined and
holomorphic. In fact, in this region, R−1 can be written as
R−1(z) = z− a(z− z0)n+1 + o(|z− z0|n+2).
Hence, we have that z0 is still a parabolic fixed point of R−1 and, in particular, that attracting
vectors of R−1 are repelling vectors of R, and vice-versa. Fixed an attracting vector v0, all
attracting and repelling vectors can be expressed as vk = eπik/n, for k ∈ {0, . . . , 2n− 1}, where
vk is an attracting vector for even k and a repelling vector for odd k.
Definition 2.21 (Directional convergence and parabolic basin). Let R be a rational map of
degree d ≥ 2 and let z0 ∈ Cˆ be a parabolic fixed point of multiplicity n + 1. For a point z ∈ Cˆ,
we say that the orbit of z converges to z0 along the direction v if Rm(z) converges to z0 and
lim
m→∞
k
√
mRm(z) = v.
Hence, we define the parabolic basin of attraction of z0 at direction v as the set of points
A(z0, v) =
{
z ∈ Cˆ : z converges to z0 along the direction v
}
.
2.2 Local theory 19
z0v0
v1
v2
v3
Figure 2.2: Attracting (blue) and repelling (red) petals of a parabolic fixed point z0 of multiplicity 3. The
attracting vectors are v0, v2 and the repelling ones v1, v3.
Definition 2.22 (Attracting and repelling petals). Let R be a rational map of degree d ≥ 2 and
let z0 ∈ Cˆ be a parabolic fixed point of multiplicity n + 1. Let v be an attracting vector of R.
We say a domain P is an attracting petal of R for z0 along v if it is forward invariant and, for
every point z ∈ Cˆ, the orbit of z enters P if and only if it converges to z0 along v.
Similarly, if R has holomorphic inverse in a domain N, and w is a repelling vector of R, we
say a domain P′ ⊆ N is a repelling petal of R along w if it is an attracting petal for R−1 along
w.
This is an essential difference with the attracting and super-attracting case. Instead of hav-
ing one basin of attraction, we defined one for each attracting vector. This idea of convergence
and repelling along a direction gets formalised by the essential result of this subsection, the
Leau-Fatou Flower Theorem.
Theorem 2.23 (Leau-Fatou Flower Theorem). Let R be a rational map of degree d ≥ 2 and z0 ∈ Cˆ
a parabolic fixed point of R with multiplicity n + 1. Let v0, . . . , v2n−1 be the attracting and repelling
vectors of R.
Then, for each attracting (repelling) vector vk, there exists a simply connected attracting (repelling)
petal Pk of R along vk, defined in the domain between the two consecutive repelling (attracting) vectors,
i.e.
Pk ⊆
{
z ∈ C : z = z0 + reiθvk, r > 0, |θ| < πn
}
.
Moreover, the union of these Pk is a neighbourhood of z0. In addition, if n ≥ 2, then each petal Pk only
intersects its two direct neighbours Pk+1 mod 2n and Pk−1 mod 2n in a simply connected domain, and for
n = 1, the corresponding repelling and attracting petals intersect each other in two simply connected
domains.
This result characterises the dynamics around parabolic fixed points. A comprehensive
proof is given in [Bea91, Theorem 6.5.4]. In this scenario, in contrast with attracting and
super-attracting fixed points, where there is only one basin of attraction, the dynamics have
a directional dependence, and we have different basins of attraction for different directions.
Furthermore, now iterates in each neighbourhood of the parabolic fixed point have different
dynamical behaviours, and so parabolic fixed points are in the Julia set.
20 Complex dynamics
2.3 Global theory: Structure of the Fatou set
We just saw that near attracting, super-attracting and parabolic fixed points we can define
certain regions, the basins of attraction, where the iterates of the rational map converge to
a constant function, the fixed point. From the definition, it is clear that these sets are open,
disjoint from other basins of attraction, and are contained in the Fatou set with boundaries in
the Julia set.
This means that the connected components of basins of attraction are in fact components of
the Fatou set. So, we will now take a step back and study these in more generality. We follow
the outline of [Ste11, Chapter 3].
Definition 2.24 (Fatou component). Let R be a rational map of degree d ≥ 2 and F its Fatou
set. We say a connected component of F is called a Fatou component.
Proposition 2.25. Let R be a rational map of degree d ≥ 2. Let U be a Fatou component of R. Then,
R(U) is a Fatou component.
Proof. This result is direct taking into account the properties of proper maps (see Appendix
A.5). If U is a Fatou component of R, since R is continuous and the Fatou set of R is completely
invariant, we must have R(U) ⊆ V for a Fatou component V. Furthermore, its boundary ∂U
must be contained in the Julia set, hence R(∂U) ⊆ ∂V, since V is contained in the Fatou set. In
conclusion, U is mapped properly onto V and so R(U) = V.
This result has great implications regarding fixed points.
Definition 2.26 (Immediate basin of attraction, Bötcher, Schröder and Leau domains). Let R be
a rational map of degree d ≥ 2 and z0 a super-attracting or attracting fixed point of R. The
immediate basin of attraction of z0, denoted by A∗(z0), is the Fatou component containing
z0. They are also called Bötcher domains if z0 is super-attracting or Schröder domains if z0 is
attracting.
If z0 is parabolic and v is an attracting vector, the immediate basin of attraction of z0 along
v, denoted by A∗(z0, v), is the Fatou domain containing attracting petals of z0 along v. They
are also called Leau domains.
Note that immediate basins of attraction must be forward invariant, since fixed points and
attracting petals are so. Furthermore, they are either simply connected or infinitely connected
(see [Mil90, Lemma 3.14]).
In the case of periodic points, if α is a super-attracting, attracting or parabolic periodic
cycle of period p of R, we can consider the immediate basin of attraction of the cycle as the
immediate basin of attraction of each point of the cycle for Rp, since F (R) = F (Rp).
Hence, we just saw that each super-attracting, attracting or parabolic fixed or periodic point
distinguishes part of the Fatou set.
In view of Proposition 2.25, we can give a first classification of general components of the
Fatou set.
2.3 Global theory: Structure of the Fatou set 21
. . .
(c)(b)(a)
Figure 2.3: Classification of invariant Fatou components. Panel (a) sketches an invariant Fatou component,
panel (b) sketches pre-periodic (light purple) and periodic domains of period 3 (blue), and panel (c) sketches (the
impossible case of) wandering domains.
Definition 2.27 (Classification of Fatou components). Let R be a rational map of degree d ≥ 2,
and let U be a Fatou component of R. We say U is
(a) periodic if Rp(U) = U for some p ≥ 1. If p = 1, we say U is an invariant Fatou
component;
(b) pre-periodic if Rm(U) is a periodic domain, for minimal m ≥ 1, but U is not periodic;
(c) wandering if Rn(U) ̸= Rm(U) for all n ̸= m.
In other words, we classify the Fatou set components according to their orbit. If it is even-
tually cyclic, we have a (pre)periodic domain, or, in a more complex scenario, if the sequence
is not cyclic we have a wandering domain. Fortunately, the following celebrated result ensures
us that this complicated case cannot happen for rational maps.
Theorem 2.28 (Sullivan’s Theorem). Let R be a rational map of degree d ≥ 2 and let U be a Fatou
component. Then, U is periodic or pre-periodic.
The proof of this theorem is nuanced and shall be read carefully. It can be found with great
detail in [Ste11, Section 3.1] and [Bea91, Chapter 8].
In view of this theorem, we know that the Fatou set of a rational map consists of different
pre-periodic domains iterating eventually into periodic domains. Furthermore, note that a
periodic domain of period p of R is an invariant Fatou component of Rp. Hence, it is enough
to give a classification of these fixdomains.
We already studied the case where the iterate family tend to a constant point, but, we must
also take into account the other possibility:
Definition 2.29 (Rotation domains). Let R be a rational map of degree d ≥ 2 and let U be an
invariant Fatou component of R. We say U is a rotation domain if none of the limit functions
of the iterate sequence is constant in U. Moreover, we say U is a
1. Siegel disc if U is simply connected and contains an indifferent fixed point,
2. Herman ring if U is doubly connected.
22 Complex dynamics
Once established these five different types of fixdomains, it is time to introduce one of the
most powerful results of complex dynamics, the Classification Theorem, proved with contribu-
tions of Fatou from 1919-20 [Fat20] and Cremer (1932).
Theorem 2.30 (Classification Theorem). Let R be a rational map of degree d ≥ 2 and U an invariant
Fatou component of R. Then, U is either a Siegel disc, a Herman Ring, a Bötcher domain, a Schröder
domain or a Leau domain.
A modern proof can be found in [Ste11, Section 3.2]. As a consequence of this result,
we know that the Fatou set must be composed of different pre-periodic domains, iterating
eventually into either an immediate basin of attraction (of a super-attracting, attracting or
parabolic fixed point) or a rotation domain. Moreover, all but Herman rings are completely
determined by fixed points.
2.4 Iteration in the unit disk and the unit circle
Having studied general properties of rational maps, we shall now consider a relevant case
for our project, iteration in the unit disk D. This will be of special interest in Chapter 3. The
key result in this subject is the Wolff-Denjoy Theorem.
Theorem 2.31 (Wolff-Denjoy). Let f : D −→ D be a holomorphic self-map of D. Then, either
(a) f is an automorphism of D, or
(b) there exists z0 ∈ D such that the iterate sequence ( f n)n converges to z0 uniformly in compact
subsets of D. In this case, we say that z0 is the Wolff-Denjoy point of f .
For the proof, we refer to [CG93, Chapter IV.3], [Ste11, Chapter 2.7] and [Aba23, Theorem
3.2.1]. Hence, any self-map of the unit disk f : D −→ D which can be extended to D has
relatively simple dynamics, since z0 is a fixed point of f , even if z0 ∈ ∂D. A more rich scenario
occurs when f cannot be extended continuously to ∂D, but this case is out of the scope of this
work.
The restriction of Blaschke products to the boundary of D defines an analytic map of the
circle. To study these objects it is useful to consider their lifts and their rotation number. We
define these concepts below, since they will be useful in Chapters 3 and 5. For the general
thery of analytic maps see, for example, [dMvS12, Chapter 1].
Definition 2.32 (Lift). Let f : ∂D −→ ∂D be a self-map of the unit circle ∂D. We say F : R −→
R is a lift of f if
e2πiF(x) = f (e2πix), for all x ∈ R.
It is directly deduced from properties of branch cuts of the argument that for differentiable
f , the lift F is differentiable. Moreover, notice that there are infinitely many lifts for every
map of the circle. In particular, F1, F2 are lifts of a differentiable map f if and only if F1(x) =
F2(x) + m for some m ∈ Z.
2.4 Iteration in the unit disk and the unit circle 23
f
f (e2πix) = e2πiF(x)
f (e2πixj)
= e2πiF(xj)
e2πixe
2πix
F
∂D ∂D
R R
e2πix
e2πix2
e2πix1
e2πix3
x1 F(x1)
F(x2)
x3
x2
x x + 1 F(x + 1)F(x)
F(x3)
Figure 2.4: Sketch of a lift of a self-map of ∂D of degree 3. Note that f covers ∂D 3 times, and hence,
F(x + 1) = F(x) + 3.
Lifts are also useful regarding proper maps. In fact, if f is a proper map of ∂D, then f has
degree k if and only if F(x+ 1) = F(x) + k, for every lift F of f . An illustration of this property
is given in Fig. 2.4. This allows us to define the rotation number.
Definition 2.33 (Rotation number). Let f be a continuous self-map of the unit circle. Let F be
a lift of f and let x ∈ R. We say the rotation number of f is the limit
ρ( f ) := lim
n→∞
Fn(x)− x
n
.
It is not hard to see that this definition does not depend on the choice of lift F or initial
point x. For a formal proof, see [dMvS12, Section 1.1].
Chapter 3
Dynamics of finite Blaschke products
In Chapter 2, we introduced a general background in complex dynamics. In this chapter,
our focus shifts to the analysis of finite Blaschke products, the core subject of this project. Recall
from Chapter 1 that a finite Blaschke product is a map of the form (1.2):
B(z) = eiθ
k
∏
l=1
z− wl
1− wlz , for θ ∈ R, wl ∈ D
Generally, the first step in characterising the dynamics of any rational map is determining
their Julia and Fatou sets. As we saw in Chapter 1, a finite Blaschke product B can be studied
both as a rational map B : Cˆ −→ Cˆ and as a self-map of the unit disk B : D −→ D. As
a consequence, their dynamics are highly influenced by both the Wolff-Denjoy Theorem 2.31
and the Classification Theorem 2.30. The results presented in this section are a reflection of
this duality, and comprise original proofs and examples from [CG93, Section III.1] and [Fle14,
Section 1.4]. Let us start with some general remarks about the dynamics of FBP.
Proposition 3.1. Let B be a finite Blaschke product of degree d ≥ 2 and let z0 be the Wolff-Denjoy point
of B.
(a) The Fatou set of B contains D and Cˆ \D. Equivalently, the Julia set of B is contained in ∂D.
(b) If z0 ∈ D, iterates of B tend to z0 in D and to 1/z0 ∈ Cˆ \D in Cˆ \D. If z0 ∈ ∂D, then iterates
of B tend to z0 in D and Cˆ \D.
In particular, both D and Cˆ \D are contained in invariant Fatou components.
(c) If z0 ∈ D, z0 and 1/z0 are fixed points of the same type.
(d) All fixed points of B different from z0 and 1/z0 must be in ∂D and must be repelling.
Proof. (a) As both D and Cˆ \D are forward invariant, iterates on these domains leave out
more than three points of Cˆ. Then, by Montel’s Theorem 2.5, they are contained in F .
(b) By the Wolff-Denjoy Theorem 2.31, all iterates of B tend to z0 in D. Let z ∈ Cˆ \D and let
us consider the reflection point zR = 1/z ∈ D. Because of the Wolff-Denjoy Theorem, the
sequence (Bn(zR))n converges to z0, and so, (1/Bn(zR))n converges to 1/z0. Now, thanks
24
25
to the reflection property of FBP, we have that Bn(z) = 1/Bn(1/z) = 1/Bn(zR) for all
n ≥ 0, and so the sequence (Bn(z))n tends to 1/z0 for z ∈ Cˆ \D.
Next, as both D and Cˆ \D are connected, they must be contained in a Fatou component.
Furthermore, as they are forward invariant, by Proposition 2.25 these Fatou components
must be sent onto themselves, i.e., they are invariant Fatou components. By Classification
Theorem 2.30, these must be immediate basins of attraction.
(c) This is a direct consequence of Proposition 1.7.
(d) Since iterates of all points in D and Cˆ \D must tend to z0 and 1/z0, respectively, there
cannot be any other fixed points in these domains. Hence, they all belong to ∂D.
Moreover, if they were super-attracting, attracting or parabolic, they would have a basin
of attraction intersecting D or Cˆ \D, which cannot happen since points there would
not tend to the Wolff-Denjoy point or its reflection. Since they cannot be irrationally
indifferent, since its derivative is real due to 1.8(b), they must be repelling.
Note that z0 must be either super-attracting, attracting, or parabolic. In fact, if z0 ∈ D, it
must be in the Fatou set of B, and so, it must be either super-attracting or attracting. If z0 ∈ ∂D,
due to Lemma 1.8(a) and Proposition 1.8(b), it must be either attracting or parabolic.
All these results indicate that a great part of the dynamics of finite Blaschke products is de-
termined by the Wolff-Denjoy point. Then, it is natural to establish the following classification:
Definition 3.2 (Classification of finite Blaschke products). Let B be a finite Blaschke product of
degree d ≥ 2, and let z0 ∈ D be its Wolff-Denjoy point. We say B is
(a) elliptic if z0 ∈ D, and so, |B′(z0)| < 1,
(b) hyperbolic if z0 ∈ ∂D and 0 < B′(z0) < 1,
(c) parabolic if z0 ∈ ∂D and B′(z0) = 1.
Furthermore, the parabolic case can be divided in two.
Proposition 3.3. (Classification of parabolic FBP) Let B be a finite Blaschke product of degree d ≥ 2,
and let z0 ∈ D be its Wolff-Denjoy point. If z0 is parabolic, then the Fatou set of B is composed of either:
(a) one Leau domain intersecting ∂D, and z0 has multiplicity 2.
(b) two Leau domains with common border ∂D, and z0 has multiplicity 3.
We say that in the first case B is simply parabolic and that in the second, B is doubly parabolic.
Proof. First, note that since z0 ∈ ∂D, we have z0 = 1/z0. Hence, as z0 is parabolic, by Proposi-
tion 3.1(b), both D and Cˆ \D are contained in Leau domains.
In the first case, both D and Cˆ \D are contained in the same Leau domain. Hence, as Leau
domains are connected, it must have some points in ∂D. Since there is only one Leau domain,
by the Leau-Fatou Flower Theorem 2.23, z0 must have multiplicity 2.
26 Dynamics of finite Blaschke products
(a)
(c)
(b)
(d)
1/z0
z0
z0
z0
z0
Figure 3.1: Classification of finite Blaschke products as a function of the Wolff-Denjoy point z0: (a) is elliptic, (b)
hyperbolic, and (c) and (d) are simply and doubly parabolic, respectively.
Alternatively, if D and Cˆ \D are contained in different Leau domains, then they must be
disjoint, and so, they cannot intersect ∂D. Hence, the Fatou set consists of only two Leau
domains, which are exactly D and Cˆ \D. By the Leau-Fatou Flower Theorem 2.23, z0 must
have multiplicity 3.
Note that this last result implies that parabolic fixed points of finite Blaschke products
cannot have multiplicity greater than 3. Hence, this classification reduces all finite Blaschke
products to only four cases, and we will be able to determine the dynamics in each one of
them.
Lemma 3.4. Let B be a finite Blaschke product of degree d ≥ 2 and J its Julia set. Then, J is either
∂D or a Cantor subset of ∂D, i.e., a closed, perfect and totally disconnected subset of ∂D.
Proof. By Proposition 3.1(a), we know that J ⊆ ∂D. Assume now that J is not the whole ∂D.
As the Fatou set F is open (Property (2) of Proposition 2.8), we can find an arc in ∂D∩F with
endpoints in the Julia set (note that J is not empty, thanks to Property (4) of Proposition 2.8).
Then, we can find a sequence (an)n ⊆ F ∩ ∂D such that an converges to some z ∈ J .
Next, by Proposition 1.8(d), for all w ∈ B−1(z), there exist open arcs1 Iz and Iw with z ∈ Iz
and w ∈ Iw such that B : Iw −→ Iz is a diffeomorphism. Since the sequence (an)n is eventually
contained in Iz, the sequence (B|−1Iz (an))n converges to w.
Since F is completely invariant, this new sequence must also be contained in the Fatou set.
Hence, points in B−1(w) are also limit points of F ∩ ∂D. By simple recursion, we have that for
all n ≥ 0, all points in B−n(z) are limit points of F ∩ ∂D.
1An open arc is a set of ∂D whose lift is an open interval of R. More details will be given in Section 4.2.2.
27
Then, if we denote (F ∩ ∂D)′ as the set of limit points of F ∩ ∂D, we have that:⋃
n≥0
B−n(z) ⊆ (F ∩ ∂D)′ =⇒ ⋃
n≥0
B−n(z) ⊆ (F ∩ ∂D)′ = (F ∩ ∂D)′,
since F ∩ ∂D contains no isolated points. Now, by Property (5) in Proposition 2.9,
J = ⋃
n≥0
B−n(z) ⊆ (F ∩ ∂D)′.
In other words, all points in the Julia set are limit points of F ∩ ∂D. Hence, there can be no
arcs in J , i.e. it is totally disconnected. By Properties (2) and (6) (in Propositions 2.8 and 2.9,
respectively), it is also closed and perfect, and so, J is a Cantor set of ∂D.
Therefore, we reduced the dynamics of finite Blaschke products to only two cases, according
to their Fatou and Julia sets. The following theorem characterise these two cases.
Theorem 3.5 (Characterisation of the dynamics of FBP). Let B be a finite Blaschke product of degree
d ≥ 2. Then, the following holds.
(a) If B is elliptic, then the Julia set of B is ∂D and the Fatou set of B consists of either two Bötcher
domains or two Schröder domains.
(b) If B is hyperbolic, then the Julia set of B is a Cantor subset of ∂D and the Fatou set of B consists
of one Schröder domain.
(c) If B is simply parabolic, the Julia set of B is a Cantor subset of ∂D and the Fatou set of B consists
of one Leau domain.
(d) If B is doubly parabolic, the Julia set of B is ∂D and the Fatou set of B consists of two Leau
domains.
Proof. Let z0 ∈ D be the Wolff-Denjoy point of B. We shall work through all cases separately.
• If B is elliptic, then iterates for all points in D tend to z0 ∈ D. Then, if any point in ∂D
was in the Fatou set, their iterates would also need to converge to z0. However, ∂D is
forward invariant, so its iterates cannot escape from ∂D. In other words, we must have
J = ∂D.
• If B is hyperbolic, then z0 ∈ ∂D is an attracting fixed point, which means that it must be
contained in the Fatou set. Then, by Lemma 3.4, J is a Cantor set of ∂D.
• The simply and doubly parabolic cases are a direct result of Proposition 3.3 and Lemma
3.4.
Finally, in order to dive into the local behaviour of finite Blaschke products, we can look
into their fixed points.
28 Dynamics of finite Blaschke products
(a)
z0
(b)
z0
(c)
z0
(d)
z0
1/z0
Figure 3.2: Characterisation of the dynamics of FBP. (a) shows the elliptic case, where the Fatou set has two
connected components, D and Cˆ \D. (b) and (c) show the hyperbolic and simply parabolic cases, respectively,
where the Fatou set has only one connected component and the Julia set is a Cantor set of ∂D. Finally, (d) shows
the doubly parabolic case, where the Fatou set has again two connected components, D and Cˆ \D.
Theorem 3.6 (Characterisation of fixed points of FBP). Let B be a finite Blaschke product of degree
d ≥ 2. Then,
(a) If B is elliptic, it has exactly d − 1 different fixed points in ∂D, one in D and another one in
Cˆ \D.
(b) If B is hyperbolic, it has exactly d + 1 different fixed points, all of which are in ∂D.
(c) If B is simply parabolic, it has exactly d different fixed points, all of which are in ∂D.
(d) If B is doubly parabolic, it has exactly d− 1 different fixed points, all of which are in ∂D.
Proof. Let us recall that fixed points are the solutions of the equation B(z) = z, which, since the
numerator and denominator of B are degree d polynomials, must have d+ 1 solutions, counted
with multiplicity. Note that a fixed point z∗ has multiplicity 1 if B′(z∗) ̸= 1 and multiplicity
n ≥ 2 if B′(z∗) = 1 and n is minimal such that B(n)(z∗) ̸= 0.
Moreover, by Proposition 3.1, the number of solutions of B(z) = z depends entirely on the
Wolff-Denjoy point z0 and its reflection 1/z0. In fact, for z0 ∈ D, we have z0 ̸= 1/z0, and both
are attracting or super-attracting. So, they have multiplicity 1 and there must be other d− 1
different fixed points in ∂D with multiplicity 1, for a total of d + 1 fixed points.
On the other hand, if z0 ∈ ∂D, now z0 = 1/z0 and everything depends on the multiplicity of
z0. If z0 has multiplicity m, then there are other d + 1−m different fixed points of multiplicity
1 in ∂D. The total of fixed points in this case is d−m + 2 in ∂D.
If we look at each case separately:
29
(a) If B is elliptic, z0 ∈ D and hence we have 1 fixed point in D, one in Cˆ \D and d− 1 in
∂D.
(b) If B is hyperbolic, B′(z0) < 1, and so, it has multiplicity 1. Then, there are exactly d + 1
fixed points in ∂D.
(c) For simply parabolic B, by Proposition 3.3, z0 has multiplicity 2 and so there are d differ-
ent fixed points in ∂D.
(d) Finally, if B is doubly parabolic, by Proposition 3.3, z0 has multiplicity 3 and so there are
d− 1 different fixed points in ∂D.
Chapter 4
Dynamics of finite Blaschke products in
the unit circle
In the previous chapter, we just saw that due to the Wolff-Denjoy Theorem, the dynamics of
finite Blaschke products in both D and Cˆ \D is simple and well-understood. For this reason,
the focus of this section is studying the behaviour of FBP in ∂D.
To do so, we first prove the existence of a conjugacy in ∂D between any elliptic or doubly
parabolic FBP and the map zd, which has some remarkable dynamical properties. Afterwards,
we find this conjugacy explicitly.
The results of this section include original proofs and background from [dMvS12, Section
II.2] and [Dev21, Chapteres 1.6 to 1.8].
4.1 Abstract conjugacy in the unit circle
We shall start by stating a well-known result in this topic, Shub’s Theorem. This result
allows us to partially characterise the behaviour of FBP in ∂D, by finding a semi-conjugacy
with a simple map, which can be taken to be zd, where d is the degree of the FBP. Before
starting and proving Shub’s Theorem, we need to introduce some general concepts about circle
maps, discussed in [dMvS12, Section II.2].
Definition 4.1 (Covering). Let f : ∂D −→ ∂D be a surjective local homeomorphism of the unit
circle. We say f is a covering of degree d if every point has exactly |d| pre-images, where d > 0
if f is orientation-preserving and d < 0 otherwise.
Due to Propositions 1.8(a) and 1.8(d), all finite Blaschke products of degree d are degree d
coverings of ∂D, with d > 0.
Definition 4.2 (Expanding map). Let f : ∂D −→ ∂D be a C1 map of ∂D. We say f is expanding
if there exist constants C > 0, λ ≥ 1 such that:∣∣( f n)′(x)∣∣ > Cλn,
for all n ≥ 1 and x ∈ D.
30
4.1 Abstract conjugacy in the unit circle 31
Notice that every expanding map of ∂D is a degree d covering for some |d| ≥ 2. We are
now ready to state Shub’s Theorem.
Theorem 4.3 (Shub’s Theorem). Let f : ∂D −→ ∂D be a degree d expanding map and g : ∂D −→
∂D a degree d covering map. Then, there exists a unique semi-conjugacy between f and g in ∂D, i.e., a
monotone, surjective and continuous map h : ∂D −→ ∂D such that
h ◦ g = f ◦ h
The idea of the proof of this theorem, which can be found in [dMvS12, Theorem II.2.1] is to
build a function space and a suitable operator in order to apply Banach Fixed Point Theorem
A.11.
Proof of Shub’s Theorem. Let us consider F, G : R −→ R lifts of f and g, respectively. Note that
as f and g are C1, we can choose F and G as diffeomorphisms.
Furthermore, since Fn is a lift of f n, we have
e2πiF
n(x) = f n(e2πix) =⇒ (Fn)′(x) = ( f n)′(e2πix)e2πi(x−Fn(x)).
And, since f is expanding, there exist C > 0,λ > 1 such that∣∣(Fn)′(x)∣∣ = ∣∣∣( f n)′(e2πix)∣∣∣ > Cλn, ∀x ∈ R.
Using this and the Inverse Function Theorem,∣∣(F−n)′(x)∣∣ = 1|(Fn)′(F−n(x))| < Kρn, ∀x ∈ R. (4.1)
with K = 1/C < ∞, ρ = 1/λ < 1.
STEP 1: Definition of the function space. Consider the space of continuous bounded
maps of R, denoted by CB(R), with the supremum distance
|ϕ1 − ϕ2| := sup
x∈R
|ϕ1(x)− ϕ2(x)| < ∞, ∀ϕ1, ϕ2 ∈ CB(R).
Endowed with this distance, CB(R) is a complete metric space. Now, let us define
E := {ϕ : R −→ R : ϕ(x + 1) = ϕ(x) + 1, ϕ continuous} .
Note that E ⊆ CB(R), since for all ϕ ∈ E , supx∈R |ϕ(x)| = supx∈[0,1] |ϕ(x)| and [0, 1] is compact.
Furthermore, E is a closed subset of CB(R). Indeed, if we have (ϕn)n ⊆ E converging to a
map ϕ : R −→ R with the supremum distance, then the convergence as maps of R is uniform.
In particular, ϕ must be continuous and satisfy ϕ(x + 1) = ϕ(x) + 1, i.e. ϕ ∈ E .
Since E is a closed subset of a complete metric space, it is also a complete metric space.
This means that we can apply Banach Fixed Point Theorem in E , as long as we find a suitable
contractible operator.
STEP 2: Define a contractible operator. Consider the operator T : E −→ E , defined as
T(ϕ) = Tϕ := F−1 ◦ ϕ ◦ G, ∀ϕ ∈ E .
32 Dynamics of finite Blaschke products in the unit circle
Observe that Tϕ is continuous, and
Tϕ(x + 1) = F−1(ϕ(G(x + 1))) = F−1(ϕ(G(x) + d)) =
= F−1(ϕ ◦ G(x) + d) = F−1 ◦ ϕ ◦ G(x) + 1 = Tϕ(x) + 1, ∀x ∈ R.
since F and G are lifts of degree d covering maps. Thus, the operator T : E −→ E is well-
defined.
Next, we see that, for n large enough, Tn is contractible. Indeed, let n ≥ 1 and ϕ1, ϕ2 ∈ E .
Then, for all x ∈ R,
|Tnϕ1(x)− Tnϕ2(x)| =
∣∣F−n(ϕ1 ◦ G(x))− F−n(ϕ2 ◦ G(x))∣∣ =
=
∣∣(F−n)′(ξ)∣∣ |ϕ1 ◦ G(x)− ϕ2 ◦ G(x)| ≤
≤ Kρn sup
x∈R
|ϕ1(x)− ϕ2(x)| = Kρn |ϕ1 − ϕ2| ,
where we applied Lagrange Mean Value Theorem for some ξ ∈ R and Eq. (4.1). Hence, since
ρ < 1, there exists n large enough so that ρn < K, and
|Tnϕ1 − Tnϕ2| < |ϕ1 − ϕ2| .
In other words, Tn is contractible for n ≥ 1 large enough. Hence, by Banach Fixed Point
Theorem A.11, there exists H ∈ E such that TH = H and, furthermore, (Tnϕ)n converges to H,
for all ϕ ∈ E . Since H ∈ E , H is continuous and satisfies H(x + 1) = H(x) + 1. Thus, H is the
lift of a continuous circle map h : ∂D −→ ∂D of degree 1, i.e. h(eix) = eiH(x). We shall see that
this map h satisfies the requirements of the theorem.
First, note that since H is a fixed point of T, we have TH = F−1 ◦ H ◦ G = H. Thus,
H ◦ G = F ◦ H. Applying the exponential function in both sides, we get h ◦ g = f ◦ h, as we
wanted to see.
Now, it is left to see that h is in fact a semi-conjugacy. We already have that it is continuous
and surjective. We claim it is monotonous. Indeed, using that H is the limit of Tnϕ as n tends
to ∞ for every map ϕ ∈ E , we have that it is, in particular, the limit of Tnid = F−n ◦ id ◦ Gn =
F−n ◦Gn. As F and G are diffeomorphisms, we have that F−n ◦Gn = Tnid must be so for every
n ≥ 1. Thus, H is the uniform limit of a sequence of strictly monotonous maps. Hence, it is
monotonous. As a straight-forward consequence, h is monotonous as well.
In conclusion, the map h : ∂D −→ ∂D is continuous, monotonous, surjective, and satisfies
h ◦ g(z) = f ◦ h(z) for all z ∈ ∂D, as we wanted to see.
Finally, the uniqueness of h comes directly from the uniqueness of H.
Now, back to our matter of study, for any finite Blaschke product B of degree d ≥ 2 we can
consider the semi-conjugacy to the expanding map f (z) = zd (note f n(z) = zd
n
, so ( f n)′(z) =
dn). The relevant question now is whether this semi-conjugacy is a topological conjugacy or
not. We have the following result, illustrated in Fig. 4.1.
Theorem 4.4 (Shub’s conjugacy). Let B be a finite Blaschke product of degree d ≥ 2 and h : ∂D −→
∂D its semi-conjugacy to f (z) = zd given by Shub’s Theorem 4.3. Then, h is a conjugacy if and only if
B is elliptic or doubly parabolic.
4.1 Abstract conjugacy in the unit circle 33
Proof. Let B be elliptic or doubly parabolic. First, let us assume h is not a conjugacy. By Shub’s
Theorem 4.3, the only possibility is that h is not strictly monotonous but just monotonous. In
other words, there exist z1, z2 ∈ ∂D such that z1 ̸= z2 and h(z1) = h(z2). Since h is monotonous,
it must also be constant in the whole arc I between z1 and z2. In other words, h(I) = p for
some p ∈ ∂D.
Second, consider any domain D such that D ∩ ∂D = I. By Theorem 3.5, since B is elliptic
or doubly parabolic, the Julia set of B, J , is equal to ∂D. Hence, by the blow-up property (8)
from 2.9, there exists n ≥ 0 such that Bn(D ∩ J ) = Bn(D ∩ ∂D) = Bn(I) = J = ∂D. Then, we
have
h ◦ gn(I) = h(∂D) = ∂D, since h is onto ∂D
h ◦ gn(I) = (h ◦ g) ◦ gn−1 = ( f ◦ h) ◦ gn−1(I) = · · · = f n ◦ h(I) = f n(p)
which is impossible, since f n(p) is a single point and cannot be the whole ∂D. In conclusion,
I Bn(I) = J = ∂D
Bn
h h
f n
h(∂D) = ∂D
h(I) = p
f n(p)
D D
DD
Figure 4.1: Contradiction of Theorem 4.4 for an elliptic or doubly parabolic finite Blaschke product.
h must be strictly monotonous. Now, since h is bijective and continuous, and ∂D is compact
and Hausdorff, it must be a homeomorphism.1 In other words, h is a topological conjugacy.
Finally, if B is hyperbolic or simply parabolic, by Theorem 3.6 it must have d+ 1 or d distinct
fixed points in ∂D, respectively. Since zd has exactly d− 1 fixed points in ∂D, there cannot be
any topological conjugacy between B and zd.
1This is a well-known result from topology, detailed in every basic course of metric spaces. For instance, see
[Sut75, Proposition 13.26].
34 Dynamics of finite Blaschke products in the unit circle
4.2 Study of Shub’s conjugacy
We just saw that the dynamics of elliptic and doubly parabolic finite Blaschke products is
relatively simple. Outside of the unit circle, all points tend to the Wolff-Denjoy point (or its
reflected point, in the elliptic case), and in the unit circle, the function is topologically conjugate
to zd where d ≥ 2 is the degree of the Blaschke product. The only loose end left to tie is the
study of this conjugacy. In this section, we find it explicitly by taking an intermediate step:
symbolic dynamics.
4.2.1 Symbolic dynamics in the unit circle
In order to characterise the topological conjugacy given in Shub’s Conjugacy Theorem 4.4
between an elliptic or doubly parabolic finite Blaschke product of degree d ≥ 2, and the map
zd, we shall introduce the framework of symbolic dynamics. More background can be found
in [KH95, Section 1.9], [Dev21, Chapters 1.6, 1.7].
Definition 4.5 (Symbolic metric space). For an integer d ≥ 2, let us define the symbolic metric
space (of d symbols) as (Σd,D), where Σd is the set of sequences of d symbols
Σd := {s = s0s1 . . . |sn ∈ {1, . . . , d} ∀n ≥ 0} ,
and D is defined as
D(s, t) :=
∞
∑
n=0
|sn − tn|
dn
, ∀s, t ∈ Σd, s = s0s1 . . . , t = t0t1 . . .
We note that D is well-defined and it is indeed a distance in Σd. Indeed, since for each
n ≥ 0, sn, tn ∈ {1, . . . , d}, their difference is smaller than d− 1. Thus, the series in the definition
of D can be bounded by (d − 1)∑n(1/dn) = d. Taking into account the properties of the
absolute value, it is straight-forward that D is a distance in Σd.
Remark 4.6. We denote the elements of Σd as s, without indicating every time that s = s0s1 . . . .
Moreover, if any sequence is eventually constant, i.e. s satisfies sn = c for all n ≥ n0 + 1, we
denote s = s0 . . . sn0 c.
Proposition 4.7. Let (Σd,D) be the symbolic metric space for d ≥ 2, and let k ≥ 0. For s, t ∈ Σd, the
distance D satisfies the following.
• If sn = tn for n ∈ {0, . . . , k}, we have D(s, t) ≤ 1dk .
• If D(s, t) < 1dk , then sn = tn for n ∈ {0, . . . , k}.
Proof. First, assume sn = tn for n ∈ {0, . . . , k}. Then,
D(s, t) =
∞
∑
n=k+1
|sn − tn|
dn
≤
∞
∑
n=k+1
(d− 1)
dn
= (d− 1)
∞
∑
n=k+1
1
dn
= (d− 1) (1/d)
k+1
1− (1/d) =
1
dk
.
4.2 Study of Shub’s conjugacy 35
Second, for the other statement, assume that there exists n0 ∈ {0, . . . , k}, such that sn0 ̸= tn0 .
We have
D(s, t) =
∞
∑
n=0
|sn − tn|
dn
≥ |sn0 − tn0 |
dn0
≥ 1
dn0
≥ 1
dk
.
Hence, conversely, if we have k ≥ 0 such that D(s, t) < 1/dk, then sn = tn for n ∈ {0, . . . , k}, as
we wanted to see.
Definition 4.8 (Shift map). Let (Σd,D) be the symbolic metric space for d ≥ 2. The shift map
of Σd is defined by
σ(s0s1s2 . . . ) = s1s2 . . . , ∀s = s0s1s2 · · · ∈ Σd.
Proposition 4.9. Let (Σd,D) be the symbolic metric space for d ≥ 2 and let σ be the shift map of Σd.
Then, σ is uniformly continuous.
Proof. Let ε > 0. Choose k ≥ 1 such that (1/dk−1) < ε. Then, for δ ≤ (1/dk+1), and s, t ∈ Σd,
we have that if D(s, t) < δ ≤ (1/dk), then sn = tn for n ∈ {0, . . . , k}, by Proposition 4.7.
In other words, the first n symbols of the sequences σ(s) and σ(t) coincide. Hence, by
Proposition 4.7, we have D(s, t) ≤ 1dk−1 < ε, so D is uniformly continuous.
4.2.2 Finding the conjugacy explicitly
Finally, we shall find Shub’s conjugacy between any elliptic or doubly parabolic finite
Blaschke product of degree d ≥ 2 and zd explicitly. To do so, we will take an intermediate
step and find a relationship between the finite Blaschke product and the shift map.
To achieve this, we divide the unit circle into d numbered regions and study which of these
are visited by any point in ∂D under iteration. Keeping track on this list of visited regions,
we are able to assign a sequence in Σd to each point of the unit circle. While it is well-known,
as mentioned in [IU23, Section 8.1], that finite Blaschke products in the unit circle admit this
construction, it has not yet been formally written.
First of all, we briefly discuss the concept of arcs in ∂D with some formality.
Definition 4.10 (Ordering in ∂D). Let p ∈ ∂D and let φp denote a branch cut of the argument
in ∂D \ {p}. We define the ordering of ∂D with origin p as the order relation ≤p defined by
φp. That is,
• z ≤p w if and only if φp(z) ≤ φp(w), for all z, w ∈ ∂D \ {p},
• p ≤p z for all z ∈ ∂D.
.
Note that the definition does not depend in the branch cut of the argument, since two
different branch cuts are equal up to additive constant.
36 Dynamics of finite Blaschke products in the unit circle
z
(z, w)p
w
p
z (w, z)p
w
p
φp
ap ap + 2π
φp(z) φp(w)
E
Figure 4.2: Sketch of the p-ordering of ∂D, where p ≤p z ≤p w, and of the arcs (z, w)p, in red, and (w, z)p, in
green.
Definition 4.11. Let p, z, w ∈ ∂D and consider the ordering of ∂D of origin p, ≤p. Let φp :
∂D −→ [ap, ap + 2π) be any branch cut of the argument with φp(p) = ap, for some ap ∈ R. Let
E(t) = eit. We define the (open) arc of endpoints z, w in the ordering of p as the arc
(z, w)p :=
{
E((z, w)), if z ≤p w
E((z, ap + 2π]) ∪ E([ap, w)), if z >p w.
We can also define the closed arc [z, w]p := (z, w)p ∪ {z, w} and the semi-open arcs [z, w)p :=
[z, w]p \ {w} and (z, w]p := [z, w]p \ {z}.
Observation 4.12. It is straight-forward to see that the definition of these arcs does not depend
in the branch cut chosen.
Back to finite Blaschke products, we shall start discussing their behaviour through the
itineraries of points. If we are working in degree d ≥ 2, it follows from Proposition 3.6 that
elliptic and doubly parabolic FBP have d− 1 ≥ 1 fixed points in ∂D, and, by Proposition 1.8(a),
each fixed point has exactly d different preimages. This is our starting point.
Definition 4.13. (Induced dynamical partition) Let B be an elliptic or doubly parabolic finite
Blaschke product of degree d ≥ 2. Let p ∈ ∂D be a fixed point of B. Let B−1(p) =
{z1 = p, z2, . . . zd} be the set of preimages of p, with z1 = p <p z2 <p z3 <p · · · <p zd.
Let us consider the set of disjoint semi-open arcs Ik = [zk, zk+1) for k ∈ {1, . . . , d− 1}, and
Id = [zd, p). We say I1, . . . , Id is the dynamical partition of ∂D induced by B with origin p.
Observation 4.14. Since, by Proposition 1.8(d), B is a local diffeomorphism in ∂D, we have that
B : Ik −→ ∂D is holomorphic and one-to-one, for k ∈ {1, . . . , d}.
This partition, dependent on B, allows us to "follow" each point through the different iter-
ates of B.
Definition 4.15 (Itinerary of a point and sequence map). Let B be an elliptic or doubly parabolic
finite Blaschke product of degree d ≥ 2, let p ∈ ∂D be a fixed point of B, and let I1, . . . , Id be
the dynamical partition of ∂D induced by B with origin p. Let Σd be the symbolic space of d
symbols.
4.2 Study of Shub’s conjugacy 37
B
B B
p = p1
z
I1
I3
I2
p2
p3
S(z)→ 1 S(z)→ 12 S(z)→ 123 S(z)→ 1233
Figure 4.3: Sketch of the itinerary of a point z ∈ ∂D. Green, red and magenta arcs represent different arcs of the
dynamical partition with origin in p.
We say that the sequence s ∈ Σd is the itinerary of z ∈ ∂D if Bn(z) ∈ Isn , for each n ≥ 0, and
define the map
S : ∂D −→ Σd
z 7−→ S(z) := z,
where z is the itinerary of z. We say S is the sequence map of B (with origin p).
Example 4.16. For instance, the itinerary of the fixed point p is p = 111 · · · = 1, since Bn(p) =
p ∈ I1 for all n ≥ 0.
Therefore, we divided ∂D into d arcs, which allow us to assign to each point on ∂D a
sequence in Σd, its itinerary. Hence, a natural question is which sequences of Σd can be realised
as itineraries, and whether two points can share the same itinerary or not. In other words, our
immediate goal is determining whether the sequence map S is one-to-one or not. For this
purpose we define the following.
Definition 4.17 (Arc of a given sequence). Let B be an elliptic or doubly parabolic finite
Blaschke product of degree d ≥ 2, let p ∈ ∂D be a fixed point of B, and let I1, . . . , Id be
the dynamical partition of ∂D induced by B with origin p. Given s0, . . . sn ∈ {1, . . . , d}, we
define the arc of sequence s0, . . . , sn as
Is0 ...sn := Is0 ∩ B−1(Is1) ∩ · · · ∩ B−n(Isn).
With this definition, z ∈ Is0 ...sn if and only if the itinerary of z starts with s0 . . . sn. It can be
seen, similarly to Obs. 4.14, that Bn+1 is holomorphic and one-to-one between any arc Is0 ...sn
and ∂D, for n ≥ 0.
The following proposition describes precisely the shape of any such arc.
Proposition 4.18. Let B be an elliptic or doubly parabolic finite Blaschke product of degree d ≥ 2, let
p ∈ ∂D be a fixed point of B, and let I1, . . . , Id be the dynamical partition of ∂D induced by B with
origin p. Then, for each finite sequence s0 . . . sn with s0, . . . , sn ∈ {1, . . . , d}, the following is satisfied.
38 Dynamics of finite Blaschke products in the unit circle
B
B
B
I1
I12
I2
Figure 4.4: Idea behind arcs of a given sequence. B maps I1 to ∂D as a diffeomorphism. Then, there is a
preimage of I2 in I1, which is I12. Hence, B maps I12 to I2 diffeomorphically, and so does B2 from I12 to ∂D.
(a) Is0 ...sn is an arc of the form [zn, wn)p, with zn, wn ∈ ∂D satisfying Bn+1(zn) = Bn+1(wn) = p.
Moreover, every point in the backward orbit of p is the endpoint of an arc of a certain sequence.
(b) Is0 ...sn ⊆ Is0 ...sn−1 . In other words, for a given sequence, its arc is nested inside the arc of their
sub-sequence.
Proof. (a) We shall prove the statement by induction on the length of the sequences, n. For
n = 0, we already saw that Ik = [z, w)p for z, w ∈ B−1(p).
Next, fix n ≥ 1 and assume it is true for all t0, . . . , tn−1 ∈ {1, . . . , d}. Let s0, . . . , sn ∈
{1, . . . , d}. We have
Is0 ...sn = Is0 ∩ B−1(Is1 ...sn)
Now, note that, by the induction hypothesis, Is1...sn = [z, w)p for z, w ∈ ∂D with Bn(z) =
Bn(w) = p.
In addition, as remarked in Obs. 4.14, B : Ik −→ ∂D is holomorphic and one-to-
one for k ∈ {1, . . . , d}. Then, the preimage B−1(Is1 ...sn) comprises exactly d semi-open
arcs [zk, wk)p, each contained in a different Ik, with B(zk) = z and B(wk) = w. Thus,
the intersection Is0 ∩ B−1(Is1 ...sn) must be [zs0 , ws0)p, where Bn+1(zs0) = Bn(z) = p and
Bn+1(ws0) = B
n(w) = p, as we wanted to see.
Conversely, if z ∈ ∂D satisfies Bn(z) = p for some n ≥ 0, then the itinerary of z is of the
form z0 . . . zn−11 for some z0, . . . , zn−1 ∈ {1, . . . , d}. Hence, z must be the endpoint of the
arc of sequence Iz0 ...zn−1 , since B
n : Iz0...zn−1 −→ ∂D is bijective.
(b) The proof can be deduced straight from Def. 4.17. We have
Is0 ...sn = Is0 ∩ B−1(Is1) ∩ · · · ∩ B−(n−1)(Isn−1) ∩ B−n(Isn) =
= Is0...sn−1 ∩ B−n(Isn) ⊆ Is0...sn−1 .
Therefore, this new concept induces smaller and smaller divisions of ∂D, each one contain-
ing points with similar itineraries. Notice that
z ∈ ∂D has itinerary s if and only if z ∈
∞⋂
n=0
Is0...sn .
4.2 Study of Shub’s conjugacy 39
In particular, if two different points share itinerary, they must be in the same infinite intersec-
tion of these kind of arcs. Hence, in order to study the possible itineraries of points of the unit
disk, it is useful to study these infinite intersections. For this purpose, it is more convenient to
work with closed sets, this means, the closure of the arcs of a given sequence.
Proposition 4.19. Let B be an elliptic or doubly parabolic finite Blaschke product of degree d ≥ 2, let
p ∈ ∂D be a fixed point of B, and let I1, . . . , Id be the dynamical partition of ∂D induced by B with
origin p. Let Σd be the symbolic space of d symbols.
Then, the intersection map
I : Σd −→ ∂D
s 7−→
∞⋂
n=0
Is0...sn ,
is well-defined, where Is0 ...sn denotes the closure of the arc of sequence s0 . . . sn, for each n ≥ 0. In
particular, each infinite intersection contains exactly one point of ∂D, and, if z ∈ ∂D has itinerary z,
we have I(z) = z.
Proof. We need to see that this infinite intersection contains exactly one element for each se-
quence. The basic idea is to apply the Nested Interval Theorem to the closed arcs of sequence.
So, let s ∈ Σd. By Proposition 4.18, for each n ≥ 0 we have Is0 ...sn = [zn, wn)p for some
zn, wn ∈ ∂D. Furthermore, if ≤p is the ordering of ∂D with origin p, then by Proposition 4.18,
we have zn ≤p zn+1 ≤p wn+1 ≤p wn for each n ≥ 0. Hence, there exist z∞ = supn zn and
w∞ = infn wn and z∞ ≤p w∞. Thus, the infinite intersection of the closed arcs of sequence is
exactly [z∞, w∞]p ̸= ∅.
To see that it contains exactly one point, we have to see z∞ = w∞. Assume, on the contrary,
z∞ <p w∞. In this case, I = (z∞, w∞)p contains infinitely many points, and we can find a
domain D ⊆ Cˆ such that D ∩ ∂D = I. Since B is elliptic or doubly parabolic, by Theorem 3.5,
the Julia set of B, J , is exactly ∂D. So, by the blow-up property (8), there exists n0 ≥ 0 such
that Bn0(D ∩ J ) = Bn0(I) = J = ∂D.
Now, we consider two points on ∂D in different arcs of the dynamical partition: y1 ∈ I1 and
y2 ∈ I2. Since Bn0(I) = ∂D, there exist x1, x2 ∈ I with Bn0(x1) = y1 ∈ I1 and Bn0(x2) = y2 ∈ I2.
Hence, the itineraries of x1 and x2 contain 1 and 2 in the n0-th position, respectively. However,
(z∞, w∞)p ⊆ [zn0 , wn0 ]p = Is0...sn0 , so all points in (z∞, w∞) must have the same itinerary. Hence,
we have reached a contradiction, implying that z∞ = w∞. Therefore, the map I is well-defined,
as we wanted to see.
For the last statement, simply note that if z ∈ ∂D has itinerary z, then
z ∈
∞⋂
n=0
Iz0...zn ⊆
∞⋂
n=0
Iz0...zn = {I(z)} ,
and thus I(z) = z.
We observe next that the intersection map I is precisely the inverse of the sequence map S.
40 Dynamics of finite Blaschke products in the unit circle
Proposition 4.20. Let B be an elliptic or doubly parabolic finite Blaschke product of degree d ≥ 2,
p ∈ ∂D a fixed point of B. Let S and I be the sequence and intersection maps of B with origin in p,
respectively. Then, they satisfy
I ◦ S = id.
Hence, the sequence map S is injective and the intersection map I is surjective.
Proof. This is a direct consequence of Proposition 4.19. In fact, let z, w ∈ ∂D. Assume S(z) =
S(w), that is, z = w. Since the intersection map I is well-defined, we have z = I(z) = I(w) =
w, as desired.
z
I1 I12
I121 z
I1213
Figure 4.5: Certain arcs of finite sub-sequences of a given sequence in Σd end up intersecting in a single point.
Then, each itinerary determines exactly one point.
Note that the previous proposition only gives us the injectivity of S and the surjectivity of
I. A natural question to ask is whether these maps are indeed one-to-one. The answer turns
out to be negative, unless restricted to appropriate subsets of Σd and ∂D. We show this in the
following proposition.
Proposition 4.21. Let B be an elliptic or doubly parabolic finite Blaschke product of degree d ≥ 2,
p ∈ ∂D a fixed point of B. Let S and I be the sequence and intersection maps of B with origin in p,
respectively. Then, the following holds.
(a) B ◦ I(s) = I ◦ σ(s) and σ(S(z)) = S(B(z)), for all s ∈ Σd and z ∈ ∂D.
(b) If s ∈ Σd satisfies Bn+1(I(s)) = p, then either s = s0 . . . sn1 or s = s0 . . . snd.
(c) S ◦ I(s) ̸= s if and only if s = s0 . . . sn−1d, for some n ≥ 0.
(d) S is not continuous at any point in the backward orbit of p.
Proof. (a) The proof is straight-forward, considering that, if z ∈ ∂D has itinerary z0z1 . . . ,
then Bn(z) has itinerary znzn+1 . . . , and that B−n(Ik) = B−n
(
Ik
)
for each n ≥ 0, k ∈
{1, . . . , d}.
4.2 Study of Shub’s conjugacy 41
(b) Simply note that fixed n ≥ 0, the semi-open arcs Is0...sn form a partition of ∂D, for all
finite sequences with s0, . . . , sn ∈ {1, . . . , d}. Hence, any point can be at most adherent to
two different Is0 ...sn . For the case of p, it is clearly in I1(n)... 1 and in Id(n)... d. Hence, I(s) = p
if and only if s = 1 or s = d. The general case is deduced directly from Propositions 4.18
and item (a).
(c) First, let s ∈ Σd be a sequence which is not any itinerary, i.e. S(I(s)) ̸= s. Then, if we
consider the point z = I(s), s cannot be the itinerary of z, which we denote by z. Hence,
there exists some n ∈ Z≥0 such that sn ̸= zn. Consequently, z ̸∈ Is0 ...sn .
Now, since z = I(s), z must be in the closure Is0 ...sn . Since Is0 ...sn and Is0...sn only differ
in one point, the endpoint, we must have that z is the endpoint of Is0 ...sn . By Proposition
4.18, this means that Bn+1(z) = p, and so, by item (b), we must have either s = s0 . . . sn1
or s = s0 . . . snd. Since one of these two must be the itinerary of z, and itineraries of
preimages of p must contain the sub-sequence 1, we conclude that the former sequence
is the itinerary of z. Hence, by hypothesis, s = s0 . . . snd.
The converse case is derived directly from item (b).
(d) Let z ∈ ∂D be such that Bn+1(z) = p, for some n ≥ 0. Then, S(z) = s0 . . . sn1 for some
s0, . . . , sn ∈ {1, . . . , d}. By Proposition 4.18, z must be the endpoint of two different arcs
of different sequences, Is0 ...sn and It0...tn , with sn ̸= tn. Hence, any neighbourhood of z, say
V, must contain points from both arcs. In other words, there exist w1, w2 ∈ V such that
w1 ∈ Is0 ...sn and w2 ∈ It0...tn . But, if we consider their sequences, we have that S(w1) starts
with s0 . . . sn and that S(w2) starts with t0 . . . tn, with tn ̸= sn. Thus, by Proposition 4.7,
D(S(w1), S(w2)) > 1/dn. In other words, S is not continuous at z.
Observe that we have just proved that, if we define the reduced set of Σd, as
Σ˜d := Σd \
{
s ∈ Σd : s = s0 . . . snd, for some n ≥ 0
}
then S ◦ I = id in this set. Hence, S and I are one-to-one, and satisfy
S ◦ B(z) = σ ◦ S(z), ∀z ∈ ∂D,
I ◦ σ(s) = B ◦ I(s), ∀s ∈ Σ˜d.
Note that S and I are not conjugacies, since they are not homeomorphisms. However, they
serve as an intermediate step for a topological conjugacy between B and zd. The key idea for
building this conjugacy is realising that zd is indeed an elliptic finite Blaschke product of degree
d with fixed point 1 (and the other (d− 1)-th roots of 1), and noting that the reduced symbolic
space does not depend in B nor p.
Theorem 4.22. Let B be an elliptic or doubly parabolic finite Blaschke product of degree d ≥ 2 and let
f (z) = zd. Let p ∈ ∂D be a fixed point of B, and consider S and I the sequence and intersection maps
of B with origin p, respectively. Similarly, consider S0 and I0 the sequence and intersection maps of f
with origin 1, respectively.
42 Dynamics of finite Blaschke products in the unit circle
B z4
p
1
z
z0
S(z) = S0(z0) = 144213 . . .
Figure 4.6: Sketch of the conjugacy between an elliptic or doubly parabolic FBP B of degree 4 and z4. Each point
z is identified with the one with the same itinerary, z0.
Then, I ◦ S0 is a topological conjugacy between B and zd. In other words, I ◦ S0 is a homeomorphism
with inverse I0 ◦ S, and
(I ◦ S0) ◦ B(z) = f ◦ (I ◦ S0)(z), ∀z ∈ ∂D.
Proof. For the sake of clarity, we shall denote ϕ = I ◦ S0. First of all, notice that the map ϕ is
well-defined. Indeed, the reduced space Σ˜d is independent on the finite Blaschke product and
the origin point. Thus, it is the same for B and for f (z) = zd, so ϕ is well-defined.
Second, by Proposition 4.20, ϕ is bijective with inverse ϕ−1 = I0 ◦ S. Hence, it is left to
study the continuity of this map and its inverse.
For this purpose, we shall lay standard notation. We denote the arc of sequence s0 . . . sn of
B with origin in p as Is0 ...sn . When working with the map f with origin 1, we refer to the arc of
such sequence as I0s0 ...sn .
It is also important to note the following.
Obs. 1. z ∈ ∂D is in the backward orbit of 1 by f if and only if ϕ(z) is in the backward
orbit of p by B.
This observation is immediate, as the itineraries of points in the backward orbit of these
fixed points are exactly those ending in 1, taking into account Proposition 4.21.
At this point, we are ready to prove that ϕ is continuous in ∂D. Thus, for z ∈ ∂D, our goal
is to prove that for every neighbourhood of ϕ(z), say V, there exists a neighbourhood of z, say
U, such that ϕ(w) ∈ V if w ∈ U. For this purpose, we consider three cases, two of which are
illustrated in Fig. 4.7.
Case 1. Let us assume z is not in the backward orbit of 1 by f . Hence, if S0(z) = z0z1 . . .
is the itinerary of z by f , we have that z is not the endpoint of any arc I0z0 ...zn for n ≥ 0, by
Proposition 4.18. Consequently, the arcs
{
I0z0...zn
}
n≥0 are a basis of neighbourhoods of z in ∂D.
Similarly, {Iz0 ...zn}n≥0 is also a basis of neighbourhoods of I ◦ S0(z) in ∂D, since I ◦ S0(z) is
not in the backward orbit of p by B, due to Obs. 1.
This way, for every neighbourhood of ϕ(z), say V, we can consider n ≥ 1 large enough so
that Iz0...zn ⊆ V. This way, choosing U = I0z0 ...zn , we have that, for w ∈ U, the itinerary of w by
f , S0(w) starts with z0 . . . zn. Hence, the point ϕ(w) = I ◦ S0(w) belongs in the arc Iz0 ...zn ⊆ V.
Thus, we have proved ϕ is continuous at z.
4.2 Study of Shub’s conjugacy 43
V
VIs0...sn
I1...1
Id...d
z
p
(a) (b)
Figure 4.7: Sketch of the basis of neighbourhoods in proof of Theorem 4.22, for the first case (a), and for the
second case (b).
Case 2. Let us assume now z = 1. In this case, 1 is exactly at one endpoint of all arcs of the
form I01...1 and I
0
d...d for any number of 1s and ds. Similarly, ϕ(1) = p is at the endpoint of all
arcs I1...1 and Id...d.
Thus, we can consider the union of these arcs, I˜0n := I01(n)... 1 ∪ I
0
d(n)... d
and I˜n := I1(n)... 1 ∪ Id(n)... d.
These conform bases of neighbourhoods of 1 and p, respectively.
Thus, for any neighbourhood of ϕ(1) = p, say V, we can find n ≥ 1 large enough so that
In ⊆ V. Thus, choosing U = I0n, we have that, for w ∈ U, the itinerary of w by f , S0(w), starts
by either n 1s or n ds. In both cases, the point ϕ(w) = I ◦ S0(w) belongs to Ik ⊆ V. Hence, ϕ is
continuous at 1.
Case 3. Finally, let us assume that z is in the backward orbit of 1, i.e. there exists n ≥ 1
such that f n(z) = 1. Note that this implies that Bn(ϕ(z)) = p.
Thus, consider any neighbourhood of ϕ(z), say V. Without loss of generality, we can change
V if needed by a smaller neighbourhood so that Bn is diffeomorphic between V and W =
Bn(V), since B is a local diffeomorphism (Proposition 1.8(d)). Moreover, W is a neighbourhood
of p = ϕ(1), as Bn(ϕ(z)) = p, and ϕ is continuous at 1. Thus, there exists some neighbourhood
of 1, U˜ such that ϕ(w) ∈ W if w ∈ U˜.
Taking into account that f n is also a local diffeomorphism and that f n(z) = 1, we have that
there exists a neighbourhood of z, say U, such that f n : U −→ U˜ is a diffeomorphism, changing
U˜ by a smaller neighbourhood of 1 if needed.
This way, we have that for w ∈ U, f n(w) ∈ U˜. By continuity of ϕ in 1, we have that
ϕ( f n(w)) ∈ W. Next, by Proposition 4.21(a), we have that
ϕ( f n(w)) = I ◦ S0( f n(w)) = I(σn(S0(w))) = Bn(I ◦ S0(w)),
where σ is the shift map of Σ˜d. Hence, Bn(ϕ(w)) ∈ W. As Bn : V −→ W is diffeomorphic, we
get that w ∈ V. In conclusion, ϕ is continuous at z, as we wanted to see.
Thus, we have that I ◦ S0 : ∂D −→ ∂D is continuous and one-to-one. Since ∂D is a
compact metric space, continuous bijective maps are homeomorphisms.2. Hence, ϕ = I ◦ S0 is
2As previously discussed, this is a well-known result from topology. See e.g. [Sut75, Theorem 13.26].
44 Dynamics of finite Blaschke products in the unit circle
a homeomorphism. Furthermore, from Proposition 4.21,
S0 ◦ f (z) = σ ◦ S0(z), ∀z ∈ ∂D,
I ◦ σ(s) = B ◦ I(s), ∀s ∈ Σ˜d.
And so, we conclude:
(I ◦ S0)(zd) = B ◦ (I ◦ S0)(z), ∀z ∈ ∂D,
as we wanted to see.
In conclusion, we have that any elliptic or doubly parabolic FBP is topologically conjugate to
zd. One might ask if the previous conjugacy has any higher degree of regularity. The answer to
this question is negative. Specifically, we have the following result by Hamilton, from [Ham96,
Theorem 1].
Theorem 4.23. Let B be an elliptic or doubly parabolic finite Blaschke product. Then, B has no ab-
solutely continuous conjugacy to another Blaschke product unless B˜ = M ◦ B ◦ M−1, with M ∈
AutH(D).
4.3 Chaos in finite Blaschke products
Now that we have proved that elliptic and doubly parabolic finite Blaschke products are
topologically conjugate to zd, and having studied this conjugacy extensively, we shall take a
brief look into the implications of this conjugacy.
Since this conjugacy is topological, an elliptic or doubly parabolic FBP and zd share the same
topological properties. This is, properties preserved by homeomorphisms, such as fixed points,
periodic points, or, notably, chaos. In this last section we will delve into this last concept. It
should be mentioned that, in the context of dynamical systems, chaos can be defined following
many different approaches. In our case, we use the convention of Devaney in [Dev21, Chapter
1.8].
Definition 4.24 (Topological transitivity). Let (X, d) be a metric space and f : X −→ X a
continuous map. We say f is topologically transitive if for any pair of open sets U, V, there
exists n ≥ 0 such that
f n(U) ∩V ̸= ∅.
Definition 4.25 (Sensitive dependence on initial conditions). Let (X, d) be a metric space and
f : X −→ X a continuous map. We say f has sensitive dependence on initial conditions if there
exists δ > 0 such that, for all x ∈ X and any neighbourhood of x, say U, there exist y ∈ U and
n ≥ 0 such that
| f n(x)− f n(y)| > δ.
Definition 4.26 (Chaos). Let (X, d) be a metric space and f : X −→ X a continuous map. We
say f is chaotic in X if periodic points of f are dense in X, f is topologically transitive and f
has sensitive dependence on initial conditions.
4.3 Chaos in finite Blaschke products 45
In our case, it is straight-forward to see that the map zd is chaotic. Indeed,
• It has sensitive dependence on initial conditions, since zd multiplies the angle between
two nearby points by d.
• It is topologically transitive, since, as we have previously argued, zd is an elliptic finite
Blaschke product. Thus, its Julia set is exactly ∂D, and the blow-up property (8) from
Proposition 2.9 directly implies topological transitivity.
• Finally, periodic points of zd are dense in ∂D. Indeed, periodic points satisfy zd
n
= z for
some n ≥ 1. Thus, they are exactly those of the form
pn,k = e2πirn,k , for rn,k =
k
dn − 1, where k ∈ Z and n ≥ 1.
Hence, given any point z = e2πiα ∈ ∂D for α ∈ R any neighbourhood of z, say U, we
can consider an contained in U, spanning an angle of ε > 0, say Aε. We claim that we
can always find a periodic point inside this arc. Indeed, if we choose n > (1/ε), then
periodic points of the form pn,k for k ∈ {1, . . . , dn − 1} are separated an angular distance
of 2πn < 2πε. Thus, there must be some periodic point pn,k inside Aε ⊆ U. In other
words, periodic points of zd are dense in ∂D.
At this point, the only loose end left to tie is whether this definition of chaos is preserved
or not through topological conjugacies. In the case of compact metric spaces, such as our case,
∂D, it is straight-forward to see that chaos is in fact a topological property. The general case is
also true, even though it is more subtle. A wider discussion can be found in [BBC+92]. Thus,
we have the following.
Theorem 4.27. Let B be an elliptic or doubly parabolic finite Blaschke product. Then, B is chaotic in
∂D.
Chapter 5
Dynamics of Generalised Blaschke
products: an example
Having studied the dynamics of finite Blaschke products in Chapter 3, we have seen that
the structure of the Fatou and Julia sets is relatively simple in those cases. The Julia set restricts
to either ∂D or a Cantor subset of ∂D, and the Fatou set is either composed of only one or two
Fatou domains, exhibiting Bötcher, Schröder or Leau domains in different cases.
Taking this into account, in this chapter we shall finish our work by showing the high
phenomenology of generalised finite Blaschke products. By only allowing the zeros of Blaschke
products to be outside the unit disk, the simple dynamical properties studied in Chapter 3
vanish and more complex structures such as rotation domains appear. Of course, this change
in behaviour can be ultimately understood with the existence of poles in D: the unit disk is no
longer forward invariant and our two main tools, Schwarz Lemma A.1 and the Wolff-Denjoy
Theorem 2.31 are no longer valid.
More background in the results of this section can be found in [dMvS12, Chapter 4].
5.1 Existence of rotation domains
Our goal in this section is to introduce some results characterising the existence of rotation
domains. Recall from section 2.3 that, given a rational map, a rotation domain is an invariant
Fatou component where the iterates do not converge to a constant map. Fatou and Julia showed
that in this case, the component must be either a topological disk (Siegel disk) or a topological
annulus (Herman ring), and in both cases, the rational map is conjugate to an irrational rigid
rotation, hence their name.
Historically, rotation domains, and in particular Herman rings, were first proved to exist in
[Her79] with a careful study of the analytic properties of diffeomorphisms of ∂D. Years later,
Shishikura introduced a new and complex method, quasi-conformal surgery, in [Shi87].
In our case, we shall follow Herman’s approach: First, we restrict our GFBP to the unit
circle and show that its iterates in this set are conjugated to an irrational rotation. Afterwards,
we extend this conjugacy to a neighbourhood of the circle and conclude it must be contained
in a rotation domain.
46
5.1 Existence of rotation domains 47
5.1.1 Conjugating to an irrational rotation
Our first step is to introduce the Herman-Yoccoz Theorem, which establishes a conjugacy
between a diffeomorphism of the unit circle and an irrational rotation. Previously, we shall
define Diophantine numbers.
Definition 5.1 (Diophantine number). We say an irrational number α ∈ R \Q is Diophantine
if there exist K > 0, µ < ∞ such that∣∣∣∣α− pq
∣∣∣∣ > Kq2+µ , for all p ∈ Z, q ∈ Z>0.
Theorem 5.2 (Herman-Yoccoz). Let f be a diffeomorphism of the unit circle ∂D. Let α = ρ( f ) be the
rotation number of f . If α is Diophantine, then f is conjugate to a rotation of angle α, ρα(z) = eiαz.
The proof of this result is well-established but involves deep mathematics. Since the pur-
pose of this chapter is just to give a short insight in the existence of rotation domains, we do
not include its proof. However, it can be found in [dMvS12, Theorem I.3.2] and [PM92, Section
3.2.b].
5.1.2 Finding a Diophantine rotation number
Herman and Yoccoz Theorem 5.2 guarantees the existence of a conjugacy from a diffeo-
morphism to an irrational rotation. However, it is necessary to find a diffeomorphism with a
Diophantine rotation number. Hence, we now give a short insight in irrational number theory
and rotation numbers.
First, in order to find a Diophantine rational number, we present one of the most important
results in number theory, Liouville’s Theorem.
Theorem 5.3 (Liouville). Let α ∈ R \Q be an algebraic irrational number of degree. In other words,
there exists an irreducible polynomial f ∈ Z[X] of degree d ≥ 2 and integer coefficients such that
f (α) = 0.
Then, there exists K > 0 such that for every p, q ∈ Z with q > 1,∣∣∣∣α− pq
∣∣∣∣ > Kqd .
The proof of this result as well as a deeper background can be found in [Sim07, Theorem
B.18.1]. In particular, it states that every algebraic irrational number is Diophantine.
Hence, we have that for every diffeomorphism of the unit circle with algebraic irrational
rotation number, we can find a conjugacy to an irrational rotation in the unit circle.
The only thing left to see is whether we can find an appropriate diffeomorphism with this
rotation number. This will be given by the following result.
48 Dynamics of Generalised Blaschke products: an example
Theorem 5.4. Let f : ∂D −→ ∂D be a homeomorphism of the unit circle. Let ρα(z) = eiαz be the
rotation of angle α. Then, the rotation function
ρ : R −→ [0, 1)
α 7−→ ρ(ρα ◦ f ),
where ρ( f ) stands for the rotation number of f , assumes every irrational value in [0, 1) exactly once.
A detailed proof of this result can be found in [dMvS12, Chapter 4].
In conclusion, due to these results, we have that, given a diffeomorphism of the unit circle
f , we can find appropriate α such that ρα ◦ f has Diophantine rotation number, and hence, is
conjugate to an irrational rotation in ∂D.
5.1.3 Extending the conjugacy
Now that we are able to conjugate an appropriate diffeomorphism to an irrational rotation
in the unit circle, the last ingredient needed to prove the existence of rotation domains is seeing
that this conjugacy can be extended to a neighbourhood of ∂D.
Theorem 5.5. Let R be a rational map of degree d ≥ 2 such that R|∂D is a diffeomorphism of the unit
circle with Diophantine rotation number.
Then, ∂D is contained in a rotation domain of R.
Proof. Let α = ρ(R|∂D) be the rotation number of R in ∂D. By the Herman-Yoccoz Theorem 5.2,
there exists a diffeomorphism φ such that φ ◦ R(z) = ρα ◦ φ(z) for z ∈ ∂D, where ρα(z) = eiαz
is the rotation of angle α.
Since φ is holomorphic, it is also analytic. Hence, it can be extended to a neighbourhood of
∂D, say U. Similarly, its holomorphic inverse φ−1 can be extended to a neighbourhood of ∂D,
V. Let Φ : U −→ U and Φ−1 : V −→ V be these extensions.
First, note that Φ−1 ◦Φ(z) = φ−1 ◦ φ(z) = z for all z ∈ ∂D. Then, by the Identity Theorem
A.3, Φ−1 is the inverse of Φ in U ∩V, so they are biholomorphisms in U ∩V.
Second, we also have that Φ ◦ B(z) = φ ◦ B(z) = ρα ◦ φ(z) = ρα ◦ Φ(z) for all z ∈ ∂D.
Hence, by the Identity Theorem Φ ◦ B = ρα ◦Φ in U ⊇ U ∩V.
This means that the conjugacy from R to ρα extends to U ∩ V, an open neighbourhood
of ∂D. In this neighbourhood we can write B(z) = Φ(eiαΦ(z)). In particular, we have that
this neighbourhood is forward invariant by B, and hence, by Montel’s Theorem 2.5, it must
be contained in the Fatou set. Since it is a domain, it must be an invariant Fatou component.
Hence, by Classification Theorem 2.30, it must either be an immediate basin of attraction or
a rotation domain. However, since B is conjugate to ρα in this domain, its iterates cannot
converge to a constant function. Hence, this neighbourhood of ∂D must be contained in a
rotation domain.
5.2 Herman rings in the family of degree 3 GFBP 49
5.2 Herman rings in the family of degree 3 GFBP
In this section, we apply the results from Section 5.1 to deduce the existence of rotation
domains in the family of generalised finite Blaschke products of degree 3 defined as
Bθ,a(z) = eiθz2
z− a
1− az , for θ, a ∈ R, a > 1.
This family comprises one of the first studied examples of Herman rings, see for instance,
[Shi87, Section 6]. Recall that we are limiting ourselves to the case a > 1, since for 0 < a < 1 is
exactly the one studied in Chapter 3, and the case a = 1 restricts to −eiθz2.
Our goal is to find appropriate θ and a such that Bθ,a is a diffeomorphism of Diophantine
rotation number in ∂D. We start by studying the critical points of Bθ,a, in other words, the
points where the derivative vanishes. Evaluating the derivative of the case θ = 0, we get
B′0,a(z) = z
−2az2 + (3+ a2)z− 2a
(1− az)2 .
And hence, the critical points in C∗ are
c± =
1
4a
(
3+ a2 ±
√
(a2 − 1)(a2 − 9)
)
.
So, for 1 < a < 3, we have that c± belong in ∂D, and for a = 3, c+ = c− = 1 a critical point of
multiplicity 2 . The case of interest is a > 3, where these two critical points are outside the unit
circle, and hence, B′0,a(z) ̸= 0 for z ∈ ∂D.
Moreover, for θ ∈ R, we have that B′θ,a(z) = B′0,a(z) ̸= 0 for z ∈ ∂D, when a > 3. Hence, we
have the following.
Proposition 5.6. Let θ ∈ R and a < 3. Then, Bθ,a is a diffeomorphism of ∂D.
Proof. Recall that Bθ,a is a proper map of degree 3. In other words, every point has exactly 3
preimages, counting with multiplicity. In particular, the unit circle ∂D must have 3 preimages,
which must also be connected compact sets.
It is direct to see that B(z) ∈ ∂D for z ∈ ∂D. Hence, we have that at least one of the three
preimages of the unit disk is the unit disk itself.
Furthermore, notice that the unit disk D contains both a zero, z = 0, and a pole of Bθ,a,
z = 1/a < 1/3. Hence, the segment joining 0 and 1/a must contain a preimage of a point of
∂D. Since preimages of ∂D must be connected, there must be a preimage of ∂D containing
this point. Note that if this new preimage intersected ∂D, the crossing points would have to
be critical points, since they would be preimages of points with multiplicity greater than 1. In
other words, there is a preimage of ∂D contained inside D. By the reflection property, there
must exist another preimage of ∂D contained in Cˆ \D.
In other words, the three preimages of ∂D are disjoint and contained in D, ∂D and Cˆ \D.
Hence, Bθ,a must cover ∂D exactly once. Since B′(z) ̸= 0 for all z ∈ ∂D, by the Inverse Function
Theorem, we have that B is a biholomorphism of the unit disk.
50 Dynamics of Generalised Blaschke products: an example
Next, we shall find appropriate θ ∈ R such that Bθ,a has Diophantine rotation number in
∂D for a certain a ∈ (0,∞). The common way to do this is numerically, computing the rotation
number for different values of θ with a certain tolerance. In fact, we can evaluate the rotation
number for different values of the two parameters, a ∈ (3,∞) and θ ∈ (0, 2π), and choose a
pair that gives the rotation number of interest. The result is shown in Fig. 5.1, where rational
Arnold tongues Tr are shown. These are the set of parameters that give a rational rotation
number r. Two irrational tongues Tα, the set of parameters giving an irrational rotation number
α, are also shown. We did evaluate these calculations, but due to the high computational power
needed to create high resolution pictures, we took Fig. 5.1 from [BFGH05]. In the same article,
a general discussion of the same family of generalised finite Blaschke products can be found.
TΦ
Figure 5.1: Tongues for the family of GFBP Bθ,a, for the parameters α = 1/a and t = θ/2π. In grey, rational
Arnold tongues for some rational numbers, alongside with the irrational tongue for θ = 5
√
2 − 1 and for Φ =
(
√
(5)− 1)/2, remarked in magenta. Figure taken from [BFGH05, Figure 1].
For historical reasons, we will use rotation number Φ = (1 +
√
5)/2 mod 1 = (
√
(5) −
1)/2, the golden ratio. We found that this rotation number occurs, for instance, with parame-
ters a = 4 and θ0 = 0.615. For these parameters, we have that Bθ0,4 is a diffeomorphism of the
unit disk with Diophantine rotation number. Hence, by Theorem 5.5, ∂D must be contained in
a rotation domain, H. Moreover, we can consider the following fact.
Proposition 5.7. For every θ ∈ R and a ∈ R, 0 is a super-attracting point of Bθ,a.
Proof. This is direct, taking into account that z2 divides Bθ,a.
Thus, 0 must have a Bötcher domain around it. This means that the rotation domain H
cannot contain the whole unit disk D. Hence, it cannot be simply connected, and so, it must
be a Herman ring. See Fig. 5.2.
5.2 Herman rings in the family of degree 3 GFBP 51
Figure 5.2: Dynamical plane of Bθ0,a, for θ0 = 2π · 0.615, a = 4. Blue and red regions indicate convergence to
the super-attracting fixed points 0 and ∞, respectively. Darker regions indicate faster convergence. Black regions
indicate the domains where the iterates do not converge to any point. In particular, the bigger annulus on the left
is a Herman ring, and other annuli correspond to its successive preimages.
Appendix A
Appendix: Tools from complex analysis
In this appendix, we shall recall some common-knowledge concepts and results from com-
plex analysis and topology that have been used throughout our work. Most of these results
can be found in any standard book of complex analysis. For instance, we refer to [Ahl66].
A.1 Basic concepts from complex analysis
Given the matter of study of this work are maps of the unit disk, the first result we shall
recall is Schwarz Lemma, detailed in [Ahl66, Theorem 4.13].
Lemma A.1 (Schwarz Lemma). Let f : D −→ D be a holomorphic self-map ofD such that f (0) = 0.
Then, | f (z)| ≤ |z| and | f ′(0)| ≤ 1.
Moreover, if either | f (z∗)| = |z∗| for some z∗ ∈ D \ {0} or | f ′(0)| = 1, then f (z) = eiθz for all
z ∈ D, for some θ ∈ R.
Next, we shall recall two basic theorems capturing the essential characteristic of holomor-
phic functions, the Maximum Modulus Principle and the Identity Theorem, which can be
found in [Ahl66, Theorem 4.12] and [Ahl66, Section 4.3.2].
Theorem A.2 (Maximum modulus Principle). Let f : Ω −→ C be a non-constant holomorphic map
defined in some domain Ω ⊆ C. Then, the function | f | has no local maxima in Ω.
Theorem A.3 (Identity Theorem). Let f , g : Ω −→ C be holomorphic functions in some domain
Ω ⊆ Cˆ. Let S ⊆ Ω be the set
S := {z ∈ Ω : f (z) = g(z)} .
If S has an accumulation point in Ω, then f (z) = g(z) for all z ∈ Ω.
As a consequence of the Identity Theorem, we shall also review the following, discussed
extensively in [Ahl66, Theorem 4.24].
52
A.2 The Riemann sphere 53
Theorem A.4 (Schwarz Reflection Principle). Let f : H+ −→ C be a holomorphic function defined
in the upper half plane,H+ = {z ∈ C : Im(z) > 0} ⊆ Cˆ. If f can be extended continuously to the real
line R, then f has a unique holomorphic extension to Cˆ, satisfying:
f (z) = f (z), for z ∈ C \H+.
Next, we shall discuss two fundamental results related to meromorphic functions and the
concept of index, detailed in [Ahl66, Theorem 4.18].
Theorem A.5 (Rouché’s Theorem). Let γ1 : R −→ Ω,γ2 : R −→ Ω be two continuous curves
defined in some domain Ω ⊆ C. Let z0 ∈ Ω. If f and g satisfy
| f (z)− g(z)| < | f (z)− z0| ,
then Ind( f , z0) = Ind(g, z0).
Theorem A.6 (Argument Principle). Let f : Ω −→ C be a meromorphic map defined in some domain
Ω ⊆ C. Let γ be any differentiable simple closed curve in Ω with interior domain D which does not
intersect any zero or pole of f . Let Z be the number of zeros of f in D counted with multiplicity and E
be the number of poles of f in D, counted taking into account its order. Then,
Ind( f ◦ γ, z) = Z− E, for all z ∈ D.
A.2 The Riemann sphere
Now, it is convenient to review the natural extension of the complex plane, the Riemann
sphere Cˆ = C∪ {∞}. Formally, the Riemann sphere is a complex Riemann manifold of dimen-
sion 1. In particular, it has a holomorphic atlas composed of two local charts, the stereographic
projections centred in opposite poles of the sphere. Each one of these charts covers the whole
Riemann sphere except one point, and the metric in the sphere is the one induced by the usual
metric in C with these charts.
Thus, in our case, whenever we are working in the Riemann sphere, we can work as we nor-
mally would in the complex plane, taking into account that rational functions can be extended
to ∞ holomorphically.
Working in the Riemann sphere, we shall also recall the concept of Möbius transformations,
detailed in [Ahl66, Section 3.3].
Definition A.7. We say a Möbius transformation is a map M : Cˆ −→ Cˆ of the form
M(z) =
az + b
cz + d
, for all z ∈ Cˆ,
for some a, b, c, d ∈ C satisfying ad− bc ̸= 0.
54 Appendix: Tools from complex analysis
Proposition A.8. Möbius transformations are the only biholomorphisms of the Riemann sphere, Cˆ.
Proposition A.9. Möbius transformations send circles in Cˆ to circles in Cˆ, taking into account that
straight lines in C are circles in Cˆ crossing the infinity point.
Finally, let us recall the Riemann mapping theorem, a key result in iteration in the unit disk.
Theorem A.10 (Riemann mapping Theorem). Let U ⊆ C be a simply connected, non-emtpy, open
set of Cˆ, such that the boundary of U contains at least 2 points.
Then, it exists a biholomorphic map between U and the unit disk D.
A.3 Operator theory
We shall also review one key result in operator theory.
Theorem A.11 (Banach Fixed Point). Let (X , d) be a complete metric space. Let T : X −→ X be
eventually contractible, i.e., for some k ≥ 1:
d(Tkx, Tky) < d(x, y), ∀x, y ∈ X
Then, there exists a unique x0 ∈ X such that T(x0) = x0. Moreover, for any x ∈ X , the sequence
(Tn(x))n converges to x0.
More background on this well-known theorem can be found in [AJS18, Theorem 1.1].
A.4 Uniformization theorem
We shall also mention the fundamental result classifying simply connected Riemann sur-
faces.
Theorem A.12 (Uniformization theorem). Let S be a simply connected Riemann surface. Then, S is
conformally equivalent to either the unit disk D, the complex plane C, or the Riemann sphere Cˆ.
The relevance of this theorem in complex dynamics is notable, since every non-simply
connected Riemann surface can be covered by a simply connected one.
A.5 Proper maps
Finally, we end this appendix by reviewing a purely topological concept which can be
studied in the framework of complex analysis, proper maps, introduced formally in [Ste11,
Section 1.2].
A.5 Proper maps 55
Definition A.13 (Proper map). Let us consider domains U, V ⊆ Cˆ. We say that a function
f : U −→ V is proper if, for every compact set K ⊆ V, the preimage f−1(K) ⊆ U is compact.
Proposition A.14. Let us consider domains U, V ⊆ Cˆ and an analytic map f : U −→ V. Then, the
following are equivalent:
(i) f is proper.
(ii) f has finite topological degree k, i.e., every point y ∈ V has exactly k preimages in U, counted
with multiplicity.
(iii) f (z) tends to ∂V as z tends to ∂U, in the following sense: if the sequence (zn)n ⊆ U is not
contained in any compact set in U, then the sequence ( f (zn))n is not contained in any compact
set in V.
For instance, rational maps of degree k are proper maps of the same degree.
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