DCDS
Discrete and continuous topological dynamics: Fields of cross sections and expansive flows
Alfonso Artigue
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 have not stable points 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 to the plane admits an expansive flow. In particular, compact surfaces do not admit expansive flows without singular points.
keywords: Topological dynamics local cross section flow homeomorphism expansive flow.
DCDS
Anomalous cw-expansive surface homeomorphisms
Alfonso Artigue
We prove that the genus two surface admits a cw-expansive homeomorphism with a fixed point whose local stable set is not locally connected.
keywords: continuum theory. Continuum-wise expansive homeomorphism derived from Anosov diffeomorphism surface homeomorphism
DCDS
Robustly N-expansive surface diffeomorphisms
Alfonso Artigue
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.
keywords: robust expansivity. hyperbolic set Expansive diffeomorphism Axiom A quasi-Anosov diffeomorphism
DCDS
Lipschitz perturbations of expansive systems
Alfonso Artigue
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.
keywords: shadowing property Expansive homeomorphisms Anosov diffeomorphisms. structural stability Lipschitz condition
DCDS
Expansive flows of surfaces
Alfonso Artigue
We prove that a flow without singular points of index zero on a compact surface is expansive if and only if the singularities are of saddle type and the union of their separatrices is dense. Moreover we show that such flows are obtained by surgery on the suspension of minimal interval exchange maps.
keywords: Dynamical systems rational billiards. interval exchange maps expansive flows flows on surfaces
DCDS
Singular cw-expansive flows
Alfonso Artigue

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.

keywords: Singular Axiom A expansive flows flows on surfaces continuum theory Lorenz attractor

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