Handel's fixed point theorem revisited

Michael Handel proved in [7] the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments [9]. These methods permitted to improve the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinityComment: Ergodic Theory and Dynamical Systems, Available on CJO 201

Topologically anosov plane homeomorphisms

This paper deals with classifying the dynamics of topologically Anosov plane homeomorphisms. We prove that a topologically Anosov homeomorphism 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 nonwan- dering 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 ≥ 0fn(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)