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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., Ministerio de Ciencia, Innovación y Universidades, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let h : R-2 -> R-2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Caratheodory's prime ends theory one can associate a rotation number, rho, to h(vertical bar U). We prove that any map in the above conditions and with rho is not an element of Q induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, rho is not an element of Q if and only if Per(h) = Fix(h) = {p}., FPU Spanish Ministry of Education, MEC, MEC, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Proyecto de innovación educativa en el que se han generado recursos digitales para su uso en el aula en varias asignaturas de Grado y Doble Grado impartidas en la Facultad de Matemáticas. El objetivo es implementar algunos ejemplos que en clase se hacen ordinaria y analógicamente en un formato digital que ayude a mejorar la visualización y comprensión de los conceptos matemáticos teóricos.

We prove the existence of minimal symplectomorphisms and strictly ergodic contactomorphisms on manifolds which admit a locally free S -action by symplectomorphisms and contactomorphisms, respectively. The proof adapts the conjugation method, introduced by Anosov and Katok, to the contact and symplectic setting.

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.

We characterize the sequences of fixed point indices {i(f(n) ,p)} n >= 1 of fixed points that are isolated as an invariant set for a continuous map f in the plane. In particular, we prove that the sequence is periodic and i(f(n) ,p) <= 1 for every n >= 0. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the 2-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree-d maps in the 2-sphere., MINECO, CNPq fellowship (Brazil), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

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., MICINN, MTM, FPU scholarship, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

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

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

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

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

Let h : R-2 -> R-2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Caratheodory's prime ends theory one can associate a rotation number, rho, to h(vertical bar U). We prove that any map in the above conditions and with rho is not an element of Q induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, rho is not an element of Q if and only if Per(h) = Fix(h) = {p}.