On Non-contractible Periodic Orbits and Bounded Deviations
Dynamical Systems
2022-01-19 v1
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.
Cite
@article{arxiv.2201.05661,
title = {On Non-contractible Periodic Orbits and Bounded Deviations},
author = {Xiao-Chuan Liu and Fabio Armando Tal},
journal= {arXiv preprint arXiv:2201.05661},
year = {2022}
}