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.