Topologically Anosov plane homeomorphisms

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 ${\cup }_{n\ge 0}{f}^n(U)={\mathbb{R}}^2$. 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.

10 pages