On stable and unstable behaviour of certain rotation segments

Autor(es)

Addas-Zanata, Salvador ; Liu, Xiao-Chuan

• Tipo de documento: Artigo Científico
• Fonte: Nonlinearity; v. 35, n. 11, p. 39-pg., 2022-11-03.

Resumo

In this paper, we study non-wandering homeomorphisms of the two-dimen-sional torus homotopic to the identity, whose rotation sets are non-trivial segments from (0, 0) to some totally irrational point (alpha, beta). We show that for any r > 1, this rotation set only appears for C ( r ) diffeomorphisms satisfying some degenerate conditions. And when such a rotation set does appear, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behaviour. More precisely, the dynamics admits bounded deviation along the direction -(alpha, beta) in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small C (0)-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism f with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by the rotation set of some nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavours interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set. (AU)

• Processo FAPESP: 18/03762-2 - Sistema dinâmicos topológicos em superfícies
• Beneficiário: Xiaochuan Liu
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado