Sobre la dinámica de homeomorfismos en torno a puntos fijos en dimensión baja

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología, leída el 30-09-2013En esta Tesis Doctoral se han estudiado dos problemas de dinámica topológica. En el primer capítulo se examina la dinámica generada por homeomorfismos del plano que conservan orientación, son disipativos y tienen un punto fijo atractivo con región de atracción U no acotada. Se analiza el caso en que el número de rotación asignado al atractor es irracional. En el segundo capítulo se estudia el índice de punto fijo para puntos fijos aislados como conjuntos invariantes para homeomorfismos que invierten orientación en R^3. Se consigue probar que dicho índice es siempre menor o igual que 1. Para la demostración se obtienen resultados sobre índice homológico de Conley discreto, del que se da una descripción completa en dimensión 1 y se aporta una nueva aproximación junto con una nueva prueba de su dualidad. Además, se caracteriza completamente las sucesiones de índices de punto en el caso anteriormente citado y también para el caso de puntos fijos no repulsores de aplicaciones continuas en el plano. In this PhD. Thesis we have studied two problems from topological dynamics. In the first chapter we study the dynamics generated by orientation-preserving planar homeomorphisms which are dissipative and have an attracting fixed point with unbounded basin of attraction U. The case in which the rotation number assigned to the attractor is irrational is studied. In the second chapter we study the fixed point index of fixed points isolated as invariant sets for orientation-reversing homeomorphisms in R^3. It is proved that the index is always less than or equal to 1. Some results about homological discrete Conley index are obtained towards the proof: a complete description in dimension 1 and a new approach together with a new proof of its duality. Furthermore, a complete characterization of the fixed point index sequence in the aforementioned case is provided, and also for the case of non-repelling fixed points for continuous maps in the planeDepto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEunpu

On the growth rate inequality for self-maps of the sphere

Let $S^m = \{x_0^2 + x_1^2 + \cdots + x_m^2 = 1\}$ and $P = \{x_0 = x_1 = 0\} \cap S^m$. Suppose that $f$ is a self--map of $S^m$ such that $f^{-1}(P) = P$ and $|\mathrm{deg}(f_{|P})| < |\mathrm{deg}(f)|$. Then, the number of fixed points of $f^n$ grows at least exponentially with base $|d| > 1$, where $d = \mathrm{deg}(f)/\mathrm{deg}(f_{|P}) \in \mathbb Z$.Comment: 10 pages, 1 figur

Uniqueness of dynamical zeta functions and symmetric products

A characterization of dynamically defined zeta functions is presented. It comprises a list of axioms, natural extension of the one which characterizes topological degree, and a uniqueness theorem. Lefschetz zeta function is the main (and proved unique) example of such zeta functions. Another interpretation of this function arises from the notion of symmetric product from which some corollaries and applications are obtained

Parrondo´s paradox for homeomorphisms

We construct two planar homeomorphisms f and g for which the origin is a globally asymptotically stable fixed point whereas for f ◦ g and g ◦ f the origin is a global repeller. Furthermore, the origin remains a global repeller for the iterated function system generated by f and g where each of the maps appears with a certain probability. This planar construction is also extended to any dimension greater than 2 and proves for first time the appearance of the Parrondo’s dynamical paradox in odd dimensions

On the components of the unstable set of isolated invariant sets

The aim of this note is to shed some light on the topological structure of the unstable set of an isolated invariant set K. We give a bound on the number of essential quasicomponents of the unstable set of K in terms of the homological Conley index of K. The proof relies on an explicit pairing between Čech homology classes and Alexander–Spanier cohomology classes that takes the form of an integral.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEMinisterio de Ciencia, Innovación y Universidadespu

About the homological discrete Conley index of isolated invariant acyclic continua

Abstract This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a locally defined homeomorphism f in R d and an acyclic continuum X, such as a cellular set or a fixed point, invariant under f and isolated. We prove that the trace of the first discrete homological Conley index of f and X is greater than or equal to -1 and describe its periodical behavior. If equality holds then the traces of the higher homological indices are 0. In the case of orientation-reversing homeomorphisms of R 3 , we obtain a characterization of the fixed point index sequence {i(f n , p)} n≥1 for a fixed point p which is isolated as an invariant set. In particular, we obtain that i(f, p) ≤ 1. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in R 3