Fully essential dynamics for area-preserving surface homeomorphisms

TL;DR

This paper introduces the concept of fully essential dynamics for area-preserving surface homeomorphisms, generalizing previous results to higher genus surfaces and analyzing the structure of their invariant sets and rotation sets.

Contribution

It defines fully essential dynamics for higher genus surfaces, extending prior toral results, and introduces homotopically bounded sets as a key tool for analyzing invariant sets.

Findings

Fully essential dynamics characterized by invariant sets with sensitive dependence on initial conditions.

Rotation sets with non-empty interior correspond to fully essential dynamics.

Invariant open sets are bounded when fixed points are inessential.

Abstract

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz-Ziemian rotation set has non-empty interior the dynamics is fully essential, and the set $C$ is (externally) sensitive on initial conditions and realizes all the rotational dynamics. As a fundamental tool we introduce the notion of homotopically bounded sets and we prove a general boundedness result for invariant open sets when the fixed point set is inessential.