Rational Mode Locking for Homeomorphisms of the 2-Torus

TL;DR

This paper investigates the stability of rational boundary points of the rotation set for homeomorphisms of the 2-torus, providing examples and theorems about how these points can or cannot be perturbed into the interior of the rotation set.

Contribution

It constructs explicit examples of perturbations that move rational boundary points into the interior or outside the rotation set, and proves conditions under which such perturbations are impossible.

Findings

Existence of $C^ abla$-diffeomorphisms with boundary points in the rotation set that can be perturbed into the interior.

In the conservative setting, only $C^0$ examples of such perturbations are possible.

Analytic area-preserving homeomorphisms cannot have boundary points of the rotation set moved into the interior by small perturbations.

Abstract

Let $f:{\rm T^2\rightarrow T^2}$ be a homeomorphism homotopic to the identity, $\widetilde{f}:{\rm I}\negthinspace {\rm R^2\rightarrow I} \negthinspace {\rm R^2}$ be a fixed lift and $\rho (\widetilde{f})$ be its rotation set, which we assume to have interior. We also assume that some rational point $(\frac pq,\frac rq)\in \partial \rho (\widetilde{f})$ and we want to understand how stable this situation is. To be more precise, we want to know if it is possible to find two different homeomorphisms, which are arbitrarily small $C^0$-perturbations of $f,$ denoted $f_1$ and $f_2,$ in a way that $(\frac pq,\frac rq)$ does not belong to the rotation set of $f_1$ and $(\frac pq,\frac rq)$ is contained in the interior of the rotation set of $f_2.$ We give two examples in this direction. The first is a $C^\infty $-diffeomorphism $f_{dissip},$ such that $(0,0)\in \partial \rho (\widetilde{f}_{dissip}),$ $f_{dissip}$ has only one fixed point with zero rotation vector and there are maps $f_1$ and $f_2$ satisfying the conditions above. The second is an area preserving version of the above, but in this conservative setting we obtain only a $C^0$ example. We also present two theorems in the opposite direction. The first says that if $f$ is area preserving and analytic, then there can not be $f_1$ and $f_2$ as above. The second result, implies that for a generic (in the sense of Brunovsky) one parameter family $% f_t:{\rm T^2\rightarrow T^2}$ of $C^1$-diffeomorphisms such that for some parameter $\overline{t},$ $\rho (\widetilde{f}_{\overline{t}})$ has interior, $(\frac pq,\frac rq) \in \partial \rho (\widetilde{f}_{\overline{t}})$ and $(\frac pq,\frac rq)\notin \rho (\widetilde{f}_t)$ for $t<\overline{t},$ then for all $t> \overline{t}$ sufficiently close to $\overline{t},$ $(\frac pq,\frac rq)\notin int(\rho ( \widetilde{f}_{\overline{t}})).$