| dc.contributor |
Jarque i Ribera, Xavier |
|
| dc.contributor |
Fagella Rabionet, Núria |
|
| dc.creator |
Espigulé Pons, Bernat |
|
| dc.date |
2019-02-28T11:57:59Z |
|
| dc.date |
2019-02-28T11:57:59Z |
|
| dc.date |
2018-06-28 |
|
| dc.date.accessioned |
2024-12-16T10:27:21Z |
|
| dc.date.available |
2024-12-16T10:27:21Z |
|
| dc.identifier |
http://hdl.handle.net/2445/129385 |
|
| dc.identifier.uri |
http://fima-docencia.ub.edu:8080/xmlui/handle/123456789/22287 |
|
| 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 |
|
| dc.description |
[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. |
|
| dc.format |
87 p. |
|
| dc.format |
application/pdf |
|
| dc.language |
eng |
|
| dc.rights |
cc-by-nc-nd (c) Bernat Espigulé Pons, 2018 |
|
| dc.rights |
http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
|
| dc.rights |
info:eu-repo/semantics/openAccess |
|
| dc.source |
Màster Oficial - Matemàtica Avançada |
|
| dc.subject |
Sistemes dinàmics complexos |
|
| dc.subject |
Fractals |
|
| dc.subject |
Treballs de fi de màster |
|
| dc.subject |
Polinomis |
|
| dc.subject |
Complex dynamical systems |
|
| dc.subject |
Fractales |
|
| dc.subject |
Master's theses |
|
| dc.subject |
Polynomials |
|
| dc.title |
Complex trees and their families of connected self-similar sets |
|
| dc.type |
info:eu-repo/semantics/masterThesis |
|