Topological horseshoes for surface homeomorphisms

Abstract

In this work, we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy, and we obtain several applications. For homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotation numbers exist for all points with nonempty ω-limit sets, and that if $\mathbb{A}$ is a generalized region of instability, then it admits a single rotation number. We also offer a new proof of a recent result of Passeggi, Potrie, and Sambarino, showing that zero entropy dissipative homeomorphisms of the annulus having a circloid as an attractor have a single rotation number.

Our work also studies homeomorphisms of the sphere without horseshoes. For these maps, we present a structure theorem in terms of fixed point free invariant subannuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most two distinct periodic orbits and that, in many cases, the restriction of the homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti et al.