Rational Polygons as Rotation Sets of Generic Homeomorphisms of the Two-Torus

TL;DR

This paper demonstrates that for a generic set of toral homeomorphisms, the rotation set is a rational polygon, and extends these results to certain diffeomorphisms, revealing diverse rotation set structures.

Contribution

It establishes the genericity of rational polygon rotation sets for toral homeomorphisms and extends the results to axiom A diffeomorphisms, also exploring minimal sets with segment rotation sets.

Findings

Open and dense set of homeomorphisms with rational polygon rotation sets

Extension of results to axiom A diffeomorphisms

Existence of minimal sets with non-trivial segment rotation sets

Abstract

We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2).