New Rotation Sets in a Family of Torus Homeomorphisms

TL;DR

This paper constructs a family of torus homeomorphisms with explicitly describable rotation sets, analyzing their bifurcations, typical behaviors, and providing new insights into rotation set structures and their generic properties.

Contribution

It introduces a new family of torus homeomorphisms with explicitly characterized rotation sets, revealing complex bifurcation phenomena and generic properties of rotation sets.

Findings

Rotation sets can be mode-locked to rational polygons on disjoint intervals.

Generic rotation sets have infinitely many extreme points accumulating on a totally irrational point.

Extreme points of rotation sets may not correspond to minimal invariant sets.

Abstract

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of $[0,1]$, consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $\rho(t)$ varies continuously with $t$, the set of extreme points of $\rho(t)$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure.