2015/08/11 by Calvez, Patrice Le, Addas-Zanata, Salvador · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1508.02597
Let f:\rm T2→ T2 be a homeomorphism homotopic to the identity, \widetildef:\rm I\negthinspace \rm R2→ I \negthinspace \rm R2 be a fixed lift and ρ(\widetildef) be its rotation set, which we assume to have interior. We also assume that some rational point (\frac pq,\frac rq)∈ ∂ ρ(\widetildef) and we want to understand how stable this situation is. To be more precise, we want to know if it is possible to find two different homeomorphisms, which are arbitrarily small C0-perturbations of f, denoted f1 and f2, in a way that (\frac pq,\frac rq) does not belong to the rotation set of f1 and (\frac pq,\frac rq) is contained in the interior of the rotation set of f2. We give two examples in this direction. The first is a C^∞ -diffeomorphism fdissip, such that (0,0)∈ ∂ ρ(\widetildefdissip), fdissip has only one fixed point with zero rotation vector and there are maps f1 and f2 satisfying the conditions above. The second is an area preserving version of the above, but in this conservative setting we obtain only a C0 example. We also present two theorems in the opposite direction. The first says that if f is area preserving and analytic, then there can not be f1 and f2 as above. The second result, implies that for a generic (in the sense of Brunovsky) one parameter family % ft:\rm T2→ T2 of C1-diffeomorphisms such that for some parameter t, ρ(\widetildef_t) has interior, (\frac pq,\frac rq) ∈ ∂ ρ(\widetildef_t) and (\frac pq,\frac rq)∉ ρ(\widetildeft) for t t sufficiently close to t, (\frac pq,\frac rq)∉ int(ρ( \widetildef_t)).