Homotopic rotation sets for higher genus surfaces

TL;DR

This paper introduces a new concept of homotopic rotation sets for higher genus surface homeomorphisms, establishing foundational properties and linking them to the existence of complex dynamical structures like horseshoes and infinitely many periodic orbits.

Contribution

It defines homotopic rotation sets for higher genus surfaces and proves their fundamental properties, connecting them to the existence of rich dynamical behaviors.

Findings

Rotation set is star-shaped.

Realization of rotation vectors by orbits and periodic orbits.

Existence of infinitely many periodic orbits when certain geodesic conditions are met.

Abstract

This paper states a definition of homotopic rotation set for higher genus surface homeomorphisms, as well as a collection of results that justify this definition. We first prove elementary results: we prove that this rotation set is star-shaped, we discuss the realisation of rotation vectors by orbits or periodic orbits and we prove the creation of new rotation vectors for some configurations.Then we use the theory developped by Le Calvez and Tal in [LCT18a] to obtain two deeper results:-- If the homotopical rotation set contains the direction of a closed geodesic which has a self-intersection, then there exists a rotational horseshoe and hence infinitely many periodic orbits in many directions.-- If the homotopical rotation set contains the directions of two closed geodesics that meet, there exists infinitely many periodic orbits in many directions.