On the triviality of flows in Alexandroff spaces
We prove that the unique possible flow in an Alexandroff $T_0$-space is the trivial one. To motivate this result, we relate Alexandroff spaces to topological hyperspaces.
math.DS↗
arXiv subjects
Francisco R. Ruiz del Portal
Publications and source records attributed to Francisco R. Ruiz del Portal.
A combinatorial description of shape theory
We give a combinatorial description of shape theory using finite topological $T_0$-spaces (finite partially ordered sets). This description may lead to a sort of computational shape theory. Then we introduce the notion of core for inverse sequences of finite spaces and prove some properties.
math.GN↗
Shape of compacta as extension of weak homotopy of finite spaces
We construct a category that classifies compact Hausdorff spaces by their shape and finite topological spaces by their weak homotopy type.
math.CT↗
Computational approximations of compact metric spaces
Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a method to approximate the homology groups of $X$ and other algebraic invariants. Finally, we study computational aspects and the implementation of this method.
math.GT↗
On some topological realizations of groups and homomorphisms
Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely $f$. In addition, realization problems involving homology, homotopy groups and groups of automorphisms are considered.
math.AT↗
Coincidence theorems for finite topological spaces
We adapt the definition of the Vietoris map to the framework of finite topological spaces and we prove some coincidence theorems. From them, we deduce a Lefschetz fixed point theorem for multivalued maps that improves recent results in the literature. Finally, it is given an application to the approximation of discrete dynamical systems in polyhedra.
math.DS↗
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 $f \circ g$ and $g \circ f$ 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.
math.DS↗
Infinite series in cohomology: attractors and Conley index
In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. Our approach exploits certain attractor-repeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if $Z$ is a metric space and $K \subseteq Z$ is a compact, global attractor for a continuous map $g \colon Z \to Z$, we show how to interpret series of the form $\sum_{j \ge 0} a_j (g^*)^j$ as endomorphisms of the cohomology group of the pair $(Z,K)$.
math.DS↗
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.
math.DS↗
Fixed point indices of planar continuous maps
We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set and continuous maps in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 1$. 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 Shub about the growth of the number of periodic orbits of degree--$d$ maps in the 2-sphere.
math.DS↗
About the homological discrete Conley index of isolated invariant acyclic continua
This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a local homeomorphism of $\mathds{R}^d$ and an invariant and isolated acyclic continuum, such as a cellular set or a fixed point. In this setting, we obtain a complete description of the first discrete homological Conley index, which is periodic, that enforces a combinatorial behavior of higher indices. As a consequence, we prove that isolated (as an invariant set) fixed points of orientation-reversing homeomorphisms of $\mathds{R}^3$ have fixed point index $\le 1$ and, as a corollary, that there are no minimal orientation-reversing homeomorphisms in $\mathds{R}^3$.
math.DS↗
Indices of the iterates of ({\Bbb R}^3)-homeomorphisms at fixed points which are isolated invariant sets
Let (U \subset {\mathbb R}^3) be an open set and (f:U \to f(U) \subset {\mathbb R}^3) be a homeomorphism. Let (p \in U) be a fixed point. It is known that, if (\{p\}) is not an isolated invariant set, the sequence of the fixed point indices of the iterates of (f) at (p), ((i(f^n,p))_{n\geq 1}), is, in general, unbounded. The main goal of this paper is to show that when (\{p\}) is an isolated invariant set, the sequence ((i(f^n,p))_{n\geq 1}) is periodic. Conversely, we show that for any periodic sequence of integers ((I_n)_{n \geq1}) satisfying Dold's necessary congruences, there exists an orientation preserving homeomorphism such that (i(f^n,p)=I_n) for every (n\geq 1). Finally we also present an application to the study of the local structure of the stable/unstable sets at (p).
math.DS↗
Indices of the iterates of $R^3$-homeomorphisms at Lyapunov stable fixed points
Given any positive sequence (\{c_n\}_{n \in {\Bbb N}}), we construct orientation preserving homeomorphisms (f:{\Bbb R}^3 \to {\Bbb R}^3) such that (Fix(f)=Per(f)=\{0\}), (0) is Lyapunov stable and (\limsup \frac{|i(f^m, 0)|}{c_m}= \infty). We will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.
math.DS↗