Topologically Anosov plane homeomorphisms

Abstract

This paper deals with classifying the dynamics of Topologically Anosov plane homeomorphisms. We prove that a Topologically Anosov home- omorphism f : R2 → R2 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 ⊂ Int(f (U )), and such that ∪n≥0f n(U ) = R2. In the general case, we prove a structure theorem for the α-limits of orbits with empty ω-limit (or the ω-limits of orbits with empty α-limit), and we show that any basin of attraction (or repulsion) must be unbounded.