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arXiv · 1805.02737

Topologically Anosov plane homeomorphisms

Gonzalo Cousillas·Jorge Groisman·Juliana Xavier
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Abstract

This paper deals with classifying the dynamics of {\it Topologically Anosov} plane homeomorphisms. We prove that a Topologically Anosov homeomorphism $f:\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwandering set of $f$ reduces to a fixed point, or if there exists an open, connected, simply connected proper subset $U$ such that $U \subset \mathrm{Int}(\overline {f(U)})$, and such that $ \cup_{n\geq 0} f^n (U)= \mathbb{R}^2$. In the general case, we prove a structure theorem for the $\alpha$-limits of orbits with empty $\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit), and we show that any basin of attraction (or repulsion) must be unbounded.

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Gonzalo Cousillas, Jorge Groisman, Juliana Xavier. 2018-05-07. Topologically Anosov plane homeomorphisms. https://doi.org/10.13140/rg.2.2.19954.40644

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