Rational mode locking for homeomorphisms of the $2$-torus

Abstract:

In this paper we consider homeomorphisms of the torus $\mathbb {R}^2/\mathbb {Z}^2$, homotopic to the identity, and their rotation sets. Let $f$ be such a homeomorphism, $\widetilde {f}:\mathbb {R}^2\to \mathbb {R}^2$ be a fixed lift and $\rho (\widetilde {f})\subset \mathbb {R}^2$ be its rotation set, which we assume to have interior. We also assume that the frontier of $\rho (\widetilde {f})$ contains a rational vector $\rho \in \mathbb {Q}^2$ 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 $f_1$ and $f_2$, arbitrarily small $C^0$-perturbations of $f$, in a way that $\rho$ does not belong to the rotation set of $\widetilde f_1$ but belongs to the interior of the rotation set of $\widetilde f_2,$ where $\widetilde f_1$ and $\widetilde f_2$ are the lifts of $f_1$ and $f_2$ that are close to $\widetilde f$. We give two examples where this happens, supposing $\rho =(0,0)$. The first one is a smooth diffeomorphism with a unique fixed point lifted to a fixed point of $\widetilde f$. The second one is an area preserving version of the first one, but in this conservative setting we only obtain a $C^0$ example. We also present two theorems in the opposite direction. The first one says that if $f$ is area preserving and analytic, we cannot find $f_1$ and $f_2$ as above. The second result, more technical, implies that the same statement holds if $f$ belongs to a generic one parameter family $(f_t)_{t\in [0,1]}$ of $C^2$-diffeomorphisms of $\mathbb {T}^2$ (in the sense of Brunovsky). In particular, lifting our family to a family $(\widetilde f_t)_{t\in [0,1]}$ of plane diffeomorphisms, one deduces that if there exists a rational vector $\rho$ and a parameter $t_*\in (0,1)$ such that $\rho (\widetilde {f}_{{t_*}})$ has non-empty interior, and $\rho \not \in \rho (\widetilde {f}_t)$ for $t<t_*$ close to $t_*$, then $\rho \not \in \mathrm {int}(\rho (\widetilde {f}_{t}))$ for all $t>t_*$ close to $t_*$. This kind of result reveals some sort of local stability of the rotation set near rational vectors of its boundary.