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

TL;DR

This paper investigates how the structure of rotation sets in generic area-preserving torus diffeomorphisms influences the existence of hyperbolic saddles and quadratic tangencies, revealing complex dynamical behaviors near boundary points of the rotation set.

Contribution

It establishes the existence of hyperbolic saddles with prescribed rotation vectors and quadratic tangencies in families of torus diffeomorphisms under specific boundary conditions of the rotation set.

Findings

Existence of hyperbolic saddles with given rotation vectors.

Presence of quadratic tangencies between stable and unstable manifolds.

Tangencies become transverse as parameters vary.

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.