Bounded deviations in higher genus I: closed geodesics

2025/11/18 by Guihéneuf, Pierre-Antoine, Tal, Fábio Armando · 2 citations
Mathematics · #37E30 #37E45 #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2511.14222

openalex publication_date 2025/11/18 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28

Abstract

This is the first article of a series of two where we study the problem of bounded deviations for homeomorphisms of closed surfaces of genus ≥ 2. This first part studies bounded deviations with respect to closed geodesics. As a byproduct of our proofs, we also get a criterion of existence of periodic orbits in terms of big deviation with respect to some closed geodesic. The combination with the second part [arXiv:2511.14226] generalises to the higher genus case most of the bounded deviations results already known for the torus.

Citations

• Surface homeomorphisms with big rotation set
• Bounded deviations in higher genus II: minimal laminations
• Geodesic tracking and the shape of ergodic rotation sets
• Hyperbolic isometries of the fine curve graph of higher genus surfaces
• Area preserving homeomorphisms of surfaces with rational rotational direction
• Parabolic isometries of the fine curve graph of the torus
• Parabolic isometries of the fine curve graph of the torus
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• On Non-contractible Periodic Orbits and Bounded Deviations
• Rotation sets and actions on curves
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• Applications of Forcing Theory to Homeomorphisms of the Closed Annulus
• Quasi-morphisms on surface diffeomorphism groups
• A condition that implies full homotopical complexity of orbits
• Topological horseshoes for surface homeomorphisms
• Fixed point sets of isotopies on surfaces
• Fully essential dynamics for area-preserving surface homeomorphisms
• Forcing theory for transverse trajectories of surface homeomorphisms
• Irrational rotation factors for conservative torus homeomorphisms
• A characterization of annularity for area-preserving toral homeomorphisms
• Rotation Sets for Maps of Tori
• Curves on 2-manifolds and isotopies

Cited by

• Surface homeomorphisms with big rotation set
• A rotational hyperbolic theory for surface homeomorphisms

Related

• Bounded deviations in higher genus II: minimal laminations
• On Non-contractible Periodic Orbits and Bounded Deviations
• Surface homeomorphisms with big rotation set
• Parabolic isometries of the fine curve graph of the torus
• Parabolic isometries of the fine curve graph of the torus