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

Part of:Smooth dynamical systems: general theoryLow-dimensional dynamical systemsTopological dynamics

Published online by Cambridge University Press:  07 October 2020

HEIDES LIMA and
PAULO VARANDAS
HEIDES LIMA
Affiliation:
Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil (e-mail: heideslima@gmail.com)
PAULO VARANDAS
Affiliation:
Departamento de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil (e-mail: paulo.varandas@ufba.br,pcvarand@gmail.com)
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Abstract

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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.

Keywords

rotation setshomeomorphisms on torihistoric behaviortopological entropymetric mean dimensiongluing orbit propertyspecification

MSC classification

Primary:37E45: Rotation numbers and vectors37B40: Topological entropy37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
Secondary:37E30: Homeomorphisms and diffeomorphisms of planes and surfaces

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 10 , October 2021, pp. 2983 - 3022
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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