28 research outputs found
Indices of the iterates of -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.Comment: 19 pages, 8 figure
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)
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
Indices of the iterates of R 3 -homeomorphisms at fixed points which are isolated invariant sets
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
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
On the triviality of flows in Alexandroff spaces
We prove that the unique possible flow in an Alexandroff T0-space is the trivial one. On the way of motivation, we relate Alexandroff spaces with topological hyperspaces
Shape compacta as extension of weak homotopy of finite spaces
We construct a category that classifes compact Hausdorff spaces by their shape and fnite topological spaces by their weak homotopy type