Topologically Anosov plane homeomorphisms

Gonzalo Cousillas, Jorge Groisman, Juliana Xavier


This paper deals with classifying the dynamics of {\it topologically Anosov} plane homeomorphisms. We prove that a topologically Anosov homeomorphism $f\colon\mathbb{R}^2 \to \mathbb{R}^2$ 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 $\overline {f(U)} \subset \rom{Int} (U)$, and such that $$ \bigcup\limits_{n\leq 0} f^n (U)= \mathbb{R}^2.$$% 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).


Topologically expansive homeomorphism; topological shadowing property; topologically Anosov plane homeomorphism; homothety

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