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arXiv · 2201.05661

On Non-contractible Periodic Orbits and Bounded Deviations

Xiao-Chuan Liu·Fabio Armando Tal
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Abstract

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic points. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic points, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or the lifted map has a single strong irrational dynamical direction.

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Xiao-Chuan Liu, Fabio Armando Tal. 2022-01-14. On Non-contractible Periodic Orbits and Bounded Deviations. https://arxiv.org/abs/2201.05661

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