2022 Realization of rotation vectors for volume preserving homeomorphisms of the torus
Paulo Varandas
Topol. Methods Nonlinear Anal. 60(2): 441-455 (2022). DOI: 10.12775/TMNA.2021.042

Abstract

In this note we study the level sets of rotation vectors for $C^0$-generic homeomorphisms in the space ${\rm Homeo}_{0,\lambda}(\mathbb T^m)$ $(m \geq 3)$ of volume preserving homeomorphisms isotopic to the identity, and contribute to the ergodic optimization of vector valued observables. It is known that such homeomorphisms satisfy the specification property and their rotation sets are polyhedrons with rational vertices and non-empty interior, and stable [3], [11], [16]. For a $C^0$-generic homeomorphism we prove uniform bounded deviations for the displacement of points in $\mathbb T^m$ in the support of any ergodic probability that generates a rotation vector in the boundary of the rotation set. As consequences, we show: (i) the support of ergodic probabilities generating rotation vectors in the boundary of rotation sets has empty interior, and (ii) weak version of Boyland's conjecture: the rotation vector of the Lebesgue measure lies in the interior of the rotation sets for a $C^0$-open and dense subset of homeomorphisms in ${\rm Homeo}_{0,\lambda}(\mathbb T^m)$.

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Paulo Varandas. "Realization of rotation vectors for volume preserving homeomorphisms of the torus." Topol. Methods Nonlinear Anal. 60 (2) 441 - 455, 2022. https://doi.org/10.12775/TMNA.2021.042

Information

Published: 2022
First available in Project Euclid: 8 September 2022

MathSciNet: MR4563242
zbMATH: 1512.37045
Digital Object Identifier: 10.12775/TMNA.2021.042

Keywords: ergodic optimization , rotation sets , specification property

Rights: Copyright © 2022 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.60 • No. 2 • 2022
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