arXiv subjects
Publications and source records attributed to Alfonso Artigue.
In this paper we study the polynomial entropy of homeomorphism on compact metric space. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater or equal than 1.
In this paper we consider expansive homeomorphisms of compact spaces with a hyperbolic metric presenting a self-similar behavior on stable and unstable sets. Several application are given related to Hausdorff dimension, entropy, intrinsically ergodic measures and the transitivity of expansive homeomorphisms with canonical coordinates.
In this paper we develop a generalization of foliated manifolds in the context of metric spaces. In particular we study dendritations of surfaces that are defined as maximal atlases of compatible upper semicontinuous local decompositions into dendrites. Applications are given in modeling stable and unstable sets of topological dynamical systems. For this purpose new forms of expansivity are defined.
In this article we construct an expansive homeomorphism of a compact three-dimensional manifold with a fixed point whose local stable set is not locally connected. This homeomorphism is obtained as a topological perturbation of a quasi-Anosov diffeomorphism that is not Anosov.
In this article we show that the three-dimensional sphere admits {transitive} expansive flows in the sense of Komuro with hyperbolic equilibrium points. The result is based on a construction that allows us to see the geodesic flow of a hyperbolic three-punctured two-dimensional sphere as the flow of a smooth vector field on the three-dimensional sphere.
We study continuum-wise expansive flows with fixed points on metric spaces and low dimensional manifolds. We give sufficient conditions for a surface flow to be singular cw-expansive and examples showing that cw-expansivity does not imply expansivity. We also construct a singular Axiom A vector field on a three-manifold being singular cw-expansive and with a Lorenz attractor and a Lorenz repeller in its non-wandering set.
In this article we give sufficient conditions for Komuro expansivity to imply the rescaled expansivity recently introduced by Wen and Wen. Also, we show that a flow on a compact metric space is expansive in the sense of Katok-Hasselblatt if and only if it is separating in the sense of Gura and the set of fixed points is open.
We show that on a totally disconnected compact metric space every separating homeomorphisms is expansive except at periodic points. We conclude that minimal separating homeomorphisms are expansive and that every separating homeomorphism has asymptotic points. We show that the only spaces admitting separating (or finite expansive) and recurrent homeomorphisms are finite sets. We apply our results to give a characterization of expansivity in terms of the expansivity of the cyclic group of powers of the homeomorphism.
We consider the problem of the observability of positively expansive maps by the time series associated to continuous real functions. For this purpose we prove a general result on the generic observability of a locally injective map of a compact metric space of finite topological dimension, extending earlier work by Gutman [6]. We apply this result to partially solve the problem of finding the minimal number of functions needed to observe a positively expansive map. We prove that two functions are necessary and sufficient for positively expansive maps on tori.
We study expansive dynamical systems from the viewpoint of general topology. We introduce the notions of orbit and refinement expansivity on topological spaces extending expansivity in the compact metric setting. Examples are given on non-Hausdorff compact spaces. Topological properties are studied in relation to separability axioms, metrizability and uniform expansivity.
It is known that if a compact metric space X admits a minimal expansive homeomorphism then X is totally disconnected. In this note we give a short proof of this result and we analyze its extension to expansive flows.
In the hyperspace of subcontinua of a compact surface we consider a second order Hausdorff distance. This metric space is compactified in such a way that the stable foliation of a pseudo-Anosov map is naturally identified with a hypercontinuum. We show that negative iterates of a stable arc converges to this hypercontinuum in the considered metric. Some dynamical properties of pseudo-Anosov maps, as topological mixing and the density of stable leaves, are generalized for cw-expansive homeomorphisms of pseudo-Anosov type on compact metric spaces.
We prove that the genus two surface admits a cw-expansive homeomorphism with a fixed point whose local stable set is not locally connected.
We give sufficient conditions for a diffeomorphism of a compact surface to be robustly $N$-expansive and cw-expansive in the $C^r$-topology. We give examples on the genus two surface showing that they need not to be Anosov diffeomorphisms. The examples are axiom A diffeomorphisms with tangencies at wandering points.
In this article we consider the general problem of translating definitions and results from the category of discrete-time dynamical systems to the category of flows. We consider the dynamics of homeomorphisms and flows on compact metric spaces, in particular Peano continua. As a translating tool, we construct continuous, symmetric and monotonous fields of local cross sections for an arbitrary flow without singular points. Next, we use this structure in the study of expansive flows on Peano continua. We show that expansive flows admit no stable point and that every point contains a non-trivial continuum in its stable set. As a corollary we obtain that no Peano continuum with an open set homeomorphic with the plane admits an expansive flow. In particular compact surface admits no expansive flow without singular points.
It is a paper about models for isolated sets and the construction of special hyperbolic Lyapunov functions. We prove that after a suitable surgery every isolated set is the intersection of an attractor and a repeller. We give linear models for attractors and repellers. With these tools we construct hyperbolic Lyapunov functions and metrics around an isolated set whose values along the orbits are catenary curves. Applications are given to expansive flows and homeomorphisms, obtaining, among other things, a hyperbolic metric on local cross sections for an arbitrary expansive flow on a compact metric space.
We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations with respect to a hyperbolic metric. We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.
In this article we consider several forms of expansivity. We introduce two new definitions related with topological dimension. We study the topology of local stable sets under cw-expansive surface homeomorphisms and expansive homeomorphisms of three-manifolds. The problem of $C^r$-stable expansivity is also considered.