Topologically Anosov plane homeomorphisms

TL;DR

This paper classifies the dynamics of Topologically Anosov plane homeomorphisms, showing they are conjugate to homotheties under certain conditions and describing the structure of their orbits and basins.

Contribution

It provides a classification of Topologically Anosov plane homeomorphisms, including conjugacy results and structure theorems for orbit limits and basins.

Findings

Homeomorphisms conjugate to homothety if they are time-one maps of flows.

Characterization of nonwandering sets as fixed points or specific subsets.

Unboundedness of basins of attraction or repulsion.

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.