Stability of the rotation set of area-preserving toral homeomorphisms

TL;DR

This paper investigates the stability of the rotation set for area-preserving toral homeomorphisms, showing that stability implies a convex polygon with rational vertices, with implications for generic dynamics and bounded deviations.

Contribution

It establishes that stable rotation sets are convex polygons with rational vertices and provides explicit bounds on these vertices, extending to both area-preserving and general cases.

Findings

Stable rotation sets are convex polygons with rational vertices.

Such homeomorphisms are generically $C^0$-dense and have bounded rotational deviations.

Explicit estimates relate the rational vertices to stability constants.

Abstract

We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are $C^0$-generic and have bounded rotational deviations (even for pseudo-orbits). The results hold both in the area-preserving setting and in the general setting. When the rotation set is stable, we give explicit estimates on the type of rationals that may appear as vertices of rotation sets in terms of the stability constants.