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On the rotation sets of generic homeomorphisms on the torus ${\mathbb T^d}$

Published online by Cambridge University Press:  07 October 2020

HEIDES LIMA
Affiliation:
Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil (e-mail: [email protected])
PAULO VARANDAS
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil (e-mail: [email protected],[email protected])

Abstract

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

Type
Original Article
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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