Surface homeomorphisms with big rotation set

TL;DR

This paper studies the structure of rotation sets for higher genus surface homeomorphisms with large rotation sets, providing bounds, invariants, and confirming a conjecture in this context.

Contribution

It proves a structure theorem for the rotation set, establishes bounds on its complexity, and confirms Boyland's conjecture for these systems.

Findings

Rotation set is a finite union of convex sets

Optimal bounds for the number of convex pieces

Existence of invariant essential open sets in certain dynamics

Abstract

This article consists in applications of [arXiv:2511.14232] in the case of homemomorphisms of higher genus surfaces whose homological rotation set is big enough -- a class of dynamics that is open. We first prove a structure theorem for the rotation set of such homeomorphisms: it is a finite union of convex sets, we get an optimal bound for the number of such pieces. This bound can be improved in the case of transitive (in this case the rotation set is convex) and non-wandering dynamics (and for such homeomorphisms we get the existence of a family of invariant essential open sets). We also get boundedness of deviations for homeomorphisms with big rotation set and some consequences of it, including a answer to Boyland's conjecture in our framework.