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Bounded deviations in higher genus I: closed geodesics

Pierre-Antoine Guihéneuf, Fábio Armando Tal

TL;DR

The paper develops a higher-genus analogue of bounded deviations by bounding orbit displacements with respect to closed geodesics on closed surfaces of genus $g\ge 2$, using Le Calvez-Brouwer forcing theory. It introduces a tracking-geodesic framework tied to ergodic measures of positive rotation speed and constructs a transverse-band apparatus to detect when large deviations force the existence of periodic points whose tracking geodesics intersect the given geodesic $\gamma$. The main result, a bounded-deviation type theorem ThBndedDevRat, follows from a case-split analysis of how lifted trajectories interact with bands and their copies, supplemented by a suite of forcing lemmas and transverse-intersection arguments. Consequences include a corollary bounding the number of lifts a lifted orbit can cross and, in turn, elliptic action on the fine curve graph in many cases, thereby connecting bounded deviations to broader dynamical-geometry structures on hyperbolic surfaces. The work lays foundational machinery for Part II (non-closed geodesics) and nodes into a broader higher-genus rotation theory by relating ergodic data to geometric tracking structures.

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 $\ge 2$. 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 [GT25] generalises to the higher genus case most of the bounded deviations results already known for the torus.

Bounded deviations in higher genus I: closed geodesics

TL;DR

The paper develops a higher-genus analogue of bounded deviations by bounding orbit displacements with respect to closed geodesics on closed surfaces of genus , using Le Calvez-Brouwer forcing theory. It introduces a tracking-geodesic framework tied to ergodic measures of positive rotation speed and constructs a transverse-band apparatus to detect when large deviations force the existence of periodic points whose tracking geodesics intersect the given geodesic . The main result, a bounded-deviation type theorem ThBndedDevRat, follows from a case-split analysis of how lifted trajectories interact with bands and their copies, supplemented by a suite of forcing lemmas and transverse-intersection arguments. Consequences include a corollary bounding the number of lifts a lifted orbit can cross and, in turn, elliptic action on the fine curve graph in many cases, thereby connecting bounded deviations to broader dynamical-geometry structures on hyperbolic surfaces. The work lays foundational machinery for Part II (non-closed geodesics) and nodes into a broader higher-genus rotation theory by relating ergodic data to geometric tracking structures.

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 . 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 [GT25] generalises to the higher genus case most of the bounded deviations results already known for the torus.

Paper Structure

This paper contains 14 sections, 45 theorems, 41 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on forcing theory
  3. Foliations and isotopies.
  4. $\mathcal{F}$-transverse paths and $\mathcal{F}$-transverse intersections.
  5. Recurrence and equivalence for $\mathcal{F}$.
  6. Accumulation property
  7. Brouwer-Le Calvez foliations and forcing theory
  8. Band defined by a transverse loop
  9. A special orbit having $\gamma$ as a tracking geodesic
  10. Transverse intersections
  11. Setting some constants
  12. First case: the trajectory stays in different copies of $\widetilde{B}$
  13. Second case: the trajectory crosses different copies of $\widetilde{B}$
  14. Final proof of Theorem \ref{['ThBndedDevRat']} and Corollary \ref{['CoroBndedDevRat']}

Key Result

Lemma 1.2

Let $\mu\in\mathcal{M}^{\mathrm{erg}}(f)$. Then there exists a constant $\vartheta_\mu\in\mathbf{R}_+$ --- called the rotation speed of $\mu$ --- such that for $\mu$-almost every point $z \in S$.

Figures (11)

  • Figure 1: Example of $\widehat{\mathcal{F}}$-transverse intersection.
  • Figure 2: proof of Lemma \ref{['LemAccumul']}. Left: construction of the path $\widehat{\alpha}'_1$. Right: final argument of the proof. Leaves of $\widehat{\mathcal{F}}$ are in orange.
  • Figure 3: Left: the objects used in Paragraph \ref{['SubSecConst']}. Right: the configuration of Proposition \ref{['PropBndedDevRatCase2']}.
  • Figure 4: The configuration of Lemma \ref{['LemConsLemYLeftRight']}.
  • Figure 5: Proof of Lemma \ref{['LemExistTrans26']} (this is the continuation of Figure \ref{['FigLemConsLemYLeftRight']}).
  • ...and 6 more figures

Theorems & Definitions (75)

  • Lemma 1.2
  • Theorem 1.3
  • Theorem A
  • Conjecture 1.4
  • Theorem 1.5
  • Definition 1.6
  • Corollary B
  • Corollary C
  • Lemma 2.1
  • Lemma 2.2
  • ...and 65 more