A consequence of the growth of rotation sets for families of diffeomorphisms of the torus

Abstract

In this paper we consider $C^\infty $-generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let $f_t:\rm{T^2\rightarrow T^2}$ be such a family, $\widetilde{f}_t:\rm I\negthinspace R^2 \rightarrow \rm I\negthinspace R^2$ be a fixed family of lifts and $\rho (\widetilde{f}_t)$ be their rotation sets, which we assume to have interior for $t$ in a certain open interval $I.$ We also assume that some rational point $(\frac pq,\frac rq)\in \partial \rho (\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ and we want to understand consequences of the following hypothesis: For all $t>\overline{t},$ $t\in I,$ $(\frac pq,\frac rq)\in int(\partial \rho (\widetilde{f}_t)).$ Under these very natural assumptions, we prove that there exists a $f_{\overline{t}}^q$-fixed hyperbolic saddle $P_{\overline{t}}$ such that its rotation vector is $(\frac pq,\frac rq)$ and, there exists a sequence $t_i>\overline{t},$ $t_i\rightarrow \overline{t},$ such that if $P_t$ is the continuation of $P_{\overline{t}}$ with the parameter, then $W^u(\widetilde{P}_{t_i})$ (the unstable manifold) has quadratic tangencies with $W^s(\widetilde{P}_{t_i})+(c,d)$ (the stable manifold translated by $(c,d)),$ where $\widetilde{P}_{t_i}$ is any lift of $P_{t_i}$ to the plane, in other words, $\widetilde{P}_{t_i}$ is a fixed point for $(\widetilde{f}_{t_i})^q-(p,r),$ and $(c,d)\neq (0,0)$ are certain integer vectors such that $W^u(\widetilde{P}_{\overline{t}})$ do not intersect $W^s(\widetilde{P}_{\overline{t}})+(c,d).$ And these tangencies become transverse as $t$ increases.