A condition that implies full homotopical complexity of orbits

TL;DR

This paper establishes conditions under which homeomorphisms of higher genus surfaces exhibit complex, rich dynamics in their universal covers, extending known results from torus homeomorphisms to more general surfaces.

Contribution

It introduces the fully essential system of curves condition and proves that such homeomorphisms have intricate dynamics, including transverse intersections of stable and unstable manifolds and a full-dimensional rotation set.

Findings

Existence of a contractible hyperbolic periodic saddle point with transverse manifolds.

Homological rotation set is a full-dimensional convex subset of a7^{2g}.

Rotation vectors in the interior are realized by invariant sets and Lebesgue measure's rotation vector lies inside the rotation set.

Abstract

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ homotopic to the identity. A set of hypotheses is presented, called fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincar\'e disk $\mathbb{D}),$ denoted $\widetilde{f},$ has complicated and rich dynamics. In this context we generalize results that hold for homeomorphisms of the torus homotopic to the identity when their rotation sets contain zero in the interior. In particular, we prove that if $f$ is a $C^{1+\epsilon }$ diffeomorphism for some $\epsilon >0$ and $\pi :\mathbb{D}\rightarrow S$ is the covering map, then there exists a contractible hyperbolic $f$-periodic saddle point $p\in S$ such that for any $\widetilde{p}\in \pi ^{-1}(p),$ $$W^u(\widetilde{p}) \pitchfork W^s(g(\widetilde{p})) $$ for all deck transformations $g\in Deck(\pi ).$ By $\pitchfork,$ we mean a topologically transverse intersection between the manifolds, see the precise definition in subsection 1.1. We also show that the homological rotation set of such a $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets, periodic orbits in the rational case, and $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. Something that implies, in case $f$ is area-preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.