On non-contractible periodic orbits and bounded deviations

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...

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Bibliographic Details
Published in:Nonlinearity 2024-07, Vol.37 (7), p.75007
Main Authors: Liu, Xiao-Chuan, Tal, Fábio Armando
Format: Article
Language:English
Subjects:
37E30
37E45
dynamical systems
rotation theory
surface dynamics
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Summary: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 orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/ad4948