2018/01/30 by Addas-Zanata, Salvador · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1801.09820
In this paper we consider C^∞ -generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let ft:\rmT2→ T2 be such a family, \widetildeft:\rm I\negthinspace R2 → \rm I\negthinspace R2 be a fixed family of lifts and ρ(\widetildeft) 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)∈ ∂ ρ(\widetildef_t) for a certain parameter t∈ I and we want to understand consequences of the following hypothesis: For all t>t, t∈ I, (\frac pq,\frac rq)∈ int(∂ ρ(\widetildeft)). Under these very natural assumptions, we prove that there exists a f_tq-fixed hyperbolic saddle P_t such that its rotation vector is (\frac pq,\frac rq) and, there exists a sequence ti>t, ti→ t, such that if Pt is the continuation of P_t with the parameter, then Wu(\widetildePti) (the unstable manifold) has quadratic tangencies with Ws(\widetildePti)+(c,d) (the stable manifold translated by (c,d)), where \widetildePti is any lift of Pti to the plane, in other words, \widetildePti is a fixed point for (\widetildefti)q-(p,r), and (c,d)≠ (0,0) are certain integer vectors such that Wu(\widetildeP_t) do not intersect Ws(\widetildeP_t)+(c,d). And these tangencies become transverse as t increases.