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 $f_t:{\mathrm{T}}^2\to {\mathrm{T}}^2$ be such a family, ${\tilde{f}}_t:\mathrm{I}{\mathrm{R}}^2\to \mathrm{I}{\mathrm{R}}^2$ be a fixed family of lifts and $\rho ({\tilde{f}}_t)$ 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 $(\frac{p}{q},\frac{r}{q})\in \partial \rho ({\tilde{f}}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ and we want to understand consequences of the following hypothesis: For all $t>\overline{t},$ $t\in I,$ $(\frac{p}{q},\frac{r}{q})\in int(\partial \rho ({\tilde{f}}_t)).$ 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 $(\frac{p}{q},\frac{r}{q})$ and, there exists a sequence $t_i>\overline{t},$ $t_i\to \overline{t},$ such that if $P_t$ is the continuation of $P_{\overline{t}}$ with the parameter, then $W^u({\tilde{P}}_{t_i})$ (the unstable manifold) has quadratic tangencies with $W^s({\tilde{P}}_{t_i})+(c,d)$ (the stable manifold translated by $(c,d)),$ where ${\tilde{P}}_{t_i}$ is any lift of $P_{t_i}$ to the plane, in other words, ${\tilde{P}}_{t_i}$ is a fixed point for $({\tilde{f}}_{t_i}{)}^q-(p,r),$ and $(c,d)\ne (0,0)$ are certain integer vectors such that $W^u({\tilde{P}}_{\overline{t}})$ do not intersect $W^s({\tilde{P}}_{\overline{t}})+(c,d).$ And these tangencies become transverse as $t$ increases.

arXiv commentto appear in Erg. Th. & Dyn. Sys