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http://hdl.handle.net/2445/129385
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DC Field | Value | Language |
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dc.contributor.advisor | Jarque i Ribera, Xavier | - |
dc.contributor.advisor | Fagella Rabionet, Núria | - |
dc.contributor.author | Espigulé Pons, Bernat | - |
dc.date.accessioned | 2019-02-28T11:57:59Z | - |
dc.date.available | 2019-02-28T11:57:59Z | - |
dc.date.issued | 2018-06-28 | - |
dc.identifier.uri | http://hdl.handle.net/2445/129385 | - |
dc.description | Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Xavier Jarque i Ribera i Núria Fagella Rabionet | ca |
dc.description.abstract | [en] The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets which includes Cantor sets, Koch curves, Lévy curves, Sierpiński gaskets, Rauzy fractals, and fractal dendrites. We note a fundamental dichotomy for n-ary complex trees that allows us to study topological changes in regions $\mathcal{R}$ where one-parameter families of connected self-similar sets are defined. Moreover, we show how to obtain these families from systems of equations encoded by tip-to-tip equivalence relations. As far as we know, these families and the sets $M , M_{0}$, and $\mathcal{K}$ that we introduce to study $\mathcal{R}$ are new. We provide a theorem, and a necessary condition, for certifying if a given tipset (self-similar set associated to a complex tree) is a fractal dendrite. We highlight a special class of totally connected tipsets that we call root-connected. And we provide a pair of theorems related to them. For a given one-parameter family we also define the set of root-connected trees $M_{0}$ which presents an asymptotic similarity between its boundary and their associated tipsets. By adapting the notion of post-critically finite self-similar set (p.c.f. for short), the open set condition, and the Hausdorff dimension, we arrive to an upper bound for the existence of p.c.f. trees in a given one-parameter family. We also provide a theorem that allows us to discard non-p.c.f. trees just by looking at some local properties. In relation to this theorem, we set a conjecture of an interesting observation that has been consistent in numerous computational experiments. The space of one-parameter families of tipset-connected complex trees has just begun to be explored. For the family $TS(z) := T \{z, 1/2, 1/4z\}$ we prove that there is a pair of regions contained in the set $\mathcal{K}$ with a piece-wise smooth boundary. We show that this piece-wise smooth boundary is a rather exceptional case by considering a closely related family, $T S(z) := T {z, -1/2, 1/4z}$. Finally we indicate how the general framework works for one-parameter families with non-fixed mirror-symmetric trees. | ca |
dc.format.extent | 87 p. | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | eng | ca |
dc.rights | cc-by-nc-nd (c) Bernat Espigulé Pons, 2018 | - |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/3.0/es/ | * |
dc.source | Màster Oficial - Matemàtica Avançada | - |
dc.subject.classification | Sistemes dinàmics complexos | cat |
dc.subject.classification | Fractals | cat |
dc.subject.classification | Treballs de fi de màster | cat |
dc.subject.classification | Polinomis | ca |
dc.subject.other | Complex dynamical systems | eng |
dc.subject.other | Fractales | eng |
dc.subject.other | Master's theses | eng |
dc.subject.other | Polynomials | en |
dc.title | Complex trees and their families of connected self-similar sets | ca |
dc.type | info:eu-repo/semantics/masterThesis | ca |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | ca |
Appears in Collections: | Màster Oficial - Matemàtica Avançada |
Files in This Item:
File | Description | Size | Format | |
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memoria.pdf | Memòria | 119.26 MB | Adobe PDF | View/Open |
This item is licensed under a Creative Commons License