Abstract ALGEBRAIC PROPERTIES OF THE LEFSCHETZ ZETA FUNCTION, PERIODIC POINTS AND TOPOLOGICAL ENTROPY

ALGEBRAIC PROPERTIES OF THE LEFSCHETZ ZETA FUNCTION, PERIODIC POINTS AND TOPOLOGICAL ENTROPY JOSEFINA CASASAYAS, JAUME LLIBRE AND ANA NUNES The Lefschetz zeta function associated to a continuous self-map f of a compact manifold is a rational function P/Q . According to the parity of the degrees of the polinomials P and Q, we analize when the set of periodic points of f is infinite and when the topological entropy is positive .


. Introduction
In dynamical systems, it is often the case that algebraic information can be used to study qualitative and quantitative properties of the system . In this paper we study the dynamical consequences of simple algebraic properties of the Lefschetz zeta function Zf (t) associated to a continuous self-map f : M -> M of a compact manifold M, which is always a rational function, Le. Zf(t) = P(t)1Q(t) where P(t) and Q(t) are polynomials . We show that there is a relation between the parity of the degrees of P(t) and Q(t) and the finiteness of the set of periodic points of f on one hand, and vanishing topological entropy on the other.
In Section 2 we shall review the definition and some basic properties of the Lefschetz zeta function associated to a self-map and, in particular, give sufficient conditions for P(t) and Q(t) to Nave a finite factorization in cyclotomic polynomials . The statements and proofs of the results are given in Section 3 .
The authors are deeply indebted to the late Professor Pere Menal who patiently taught us the algebra that we came upon trying to prove the results of this paper and the ones of [CLN] .

Lefschetz zeta function and cyclotomic polynomials
Given a continuous self-map f of a compact manifold M  where jk = dimQHk(M ; Q), see [F1] for more details.
The following theorem is due to Fried (see Theorem 6 of [F2]) .
Theorem 2 .1 . Let M be a compact manifold and f : M ---> Int (M) a C1 map with finitely many periodic points . Then Zf(t) has a finite factorization in terms of the form (1 f tr)f1 with r E N.
As usual, we shall use the notation cn(t) for the n-th cyclotomic polynomial defined by Notice that all the zeros of cn(t) are roots of unity. A proof of t11e next proposition may be found in [L] .
Proposition 2.2. Let~be a primitive n-th root of unity and P(t) a polynomial with rational coefficients . If P(~) = 0 then cn(t)jP(t) .
Clearly, the degree W(n) Of cn (t) verifies n = 1: W(d) din and so W(n) is the Euler function, which may be computed through where n = pi' . . . p" is the prime decomposition of n. We remark that W(n) is even for n > 2, and cp(1) = cp(2) = 1 .
Proposition 2.3. Let M be a compact manifold and f : M --> Int (M) a Cl map with finitely many periodic points . Then Zf (t) may be written in the form Proof-From Theorem 2.1, Zf(t) is a rational function P(t)/Q(t) and all the roots of P(t) and Q(t) are roots of unity . Hence, by Proposition 2.2, the result follows. Now we shall show that the previous result holds for continuous surface maps with zero topological entropy. As usual we denote by h(f) the topological entropy of a continuous map f defined on a compact metric space . ProoL If we prove that all the non-zero eigenvalues Of f*k , k = 0, 1, 2, are roots of unity, then the result follows from (2.1) and Proposition 2.2 .
Since f*o is the identity, the unique eigenvalue of f*o is 1. Let us consider now f*2 . If M is non-orientable, then 0 is the only eigenvalue of f* 2 because H2 (M; Q) zz~0 . If M is orientable, f* 2(1) is the degree D of f. From [MP] we know that if 1 D1 > 1 then h(f) >_ log 1D1 . Hence, if h(f) = 0, ¡Di <_ 1 and so the only possible eigenvalues for f*2 are -1,0 and 1.
Finally, consider f*1 . By Theorem 2 of [M], if h(f) = 0 then all the eigenvalues A of f*1 satisfy JAJ < 1 . We claim that every non-zero eigenvalue A of f*1 has modulus 1 . Let A1, . . . . Ak be the non-zero eigenvalues of f* 1 . Then Moreover (2.2) must be a polynomial with intéger coefficients, because f*1 FIk is an integral matrix . Hence, in particular, ¡=l Az must belong to Z. Therefore, JIk 1 J> ,zJ > 1, and the claim follows because JA¡J < 1.
In short, (2.2) is a polynomial with integer coefficients, constant term ±1 and all its roots have modulus 1. By a standard result in algebra (see Lemma 6 .1 of [W]) the proposition follows.

Main results
Let R(t) be a polynomial. We define R* (t) by where a and Q are non-negative integers such that 1-t and 1 + t do not divide R*(t) .
Theorem 3.1 . Let f : M -Int (M) be a C1 map on the compact manifold M and let P(t)/Q(t) be its Lefschetz zeta function. If P* (t) or Q* (t) has odd degree, then f has infinitely many periodic points .
Proof. Suppose that f has finitely many periodic points . By Proposition 2.3, P* (t) and Q* (t) factorize into a product of cyclotomic polynomials c,,(t) with m > 2 . So the degrees of P* (t) and Q* (t) are even, and the theorem follows .
Theorem 3.2 . Let f : M ---> M be a continuous map on the compact connected surface M and let P(t)/Q(t) be its Lefschetz zeta function . If P* (t) or Q* (t) has odd degree, then f has positive topological entropy.
Proof. Suppose that f has zero topological entropy. Then the theorem follows from Proposition 2.4 using the same arguments of the proof of Theorem 3.1. Theorem 3.3. Let T' be the n-dimensional torus and let f : Tn -T' be a continuous map with associated Lefschetz zeta function P(t)/ Q(t) . If P* (t) or Q* (t) has odd degree, then f has infinitely many (least) periods.
Proof. Suppose that f has finitely many periods . Then, by Theorem 4 .1 and the proof of Corollary 4 .2 of [ABLSS] it follows that all the non-zero eigenvalues of f* 1 are roots of unity.
By Kunneth's formula the whole homology vector space of Tn is given as a tensor product of H* (S1 ; Q) and so one obtains all the eigenvalues of f* k as a product of the eigenvalues of f*i . Since all the roots of P(t) and Q(t) are roots of unity, by Proposition 2 .2, P(t) and Q(t) have a finite factorization in cyclotomic polynomials . Therefore, the result follows as in the proof of Theorem 3.1 . Let P(t)/Q(t) be the Lefschetz zeta function of f . If P*(t) orQ*(t) has odd degree, then f has positive topological entropy.
Proof. Suppose that h(f) = 0. By the same argument of the proof of Proposition 2 .4 we have that all the non-zero eigenvalues of f*o and f*i are roots of unity. So, by hypothesis, P(t) and Q(t) have a finite factorization in cyclotomic polynomials . Hence, the result follows as in the proof of Theorem 3.1 . a We remark that the hypothesis that for k = 2, . . ., n all the eigenvalues Of f*k can be obtained as products of the eigenvalues of f*1 holds for continuous self-maps of Tn, and also for continuous maps of many Eilenberg-Mac Lane spaces.