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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 $d = \mathrm{deg}(f)/\mathrm{deg}(f_{|P}) \in \mathbb Z$., 10 pages, 1 figure

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.

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary., Comment: 9 pages, 1 figure

En 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 plane