Generic Rotation Sets in Hyperbolic Surfaces

TL;DR

This paper demonstrates that for generic surface homeomorphisms homotopic to the identity on hyperbolic surfaces, the rotation set is composed of a bounded number of convex sets, with examples confirming the optimality of this bound.

Contribution

It establishes an upper bound on the number of convex sets forming the rotation set for generic hyperbolic surface homeomorphisms and provides examples showing this bound is sharp.

Findings

Rotation set is a union of at most 2^{5g-3} convex sets.

Examples confirm the sharpness of the bound.

Results apply to generic homeomorphisms homotopic to the identity.

Abstract

We show that for generic homeomorphisms homotopic to the identity in a closed and oriented surface of genus $g>1$, the rotation set is given by a union of at most $2^{5g-3}$ convex sets. Examples showing the sharpness for this asymptotic order are provided.