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A rotational hyperbolic theory for surface homeomorphisms

Pierre-Antoine Guihéneuf

TL;DR

This work builds a rotationally hyperbolic framework for surface homeomorphisms by translating rotational data from ergodic measures into hyperbolic-like rotational classes, then organizing these through forcing theory to produce a network of rotational horseshoes. It defines five heteroclinic-type relations, proves them equivalent, and encodes the global rotational structure in graphs $G$ and $\mathcal{T}$ to describe how rotation vectors assemble into $rot(f)$. The theory yields realizability and convex-structure results for rotation sets, and constructs invariant open sets tied to the class graph, laying the groundwork for a rotational Axiom A-type viewpoint with a companion applications paper for big rotation sets.

Abstract

We develop a rotational hyperbolic theory for surface homeomorphisms. We use the equivalence relation on ergodic measures that have nontrivial rotational behaviour defined in arXiv:2312.06249 to define a rotational counterpart of homoclinic classes. These allows to produce a network of horseshoes representing the whole rotational behaviour f the homeomorphism. We also study the counterpart of heteroclinic connections and give 5 different characterizations of such connections. The main technical tool is the forcing theory of Le Calvez and Tal arXiv:1503.09127, arXiv:1803.04557, and in particular a result of creation of periodic points that can also be seen as a statement of homotopically bounded deviations [GT25a]. This theoretical article is followed by a paper focused of some applications of it to the case of homeomorphisms with big rotation set [Gui25].

A rotational hyperbolic theory for surface homeomorphisms

TL;DR

This work builds a rotationally hyperbolic framework for surface homeomorphisms by translating rotational data from ergodic measures into hyperbolic-like rotational classes, then organizing these through forcing theory to produce a network of rotational horseshoes. It defines five heteroclinic-type relations, proves them equivalent, and encodes the global rotational structure in graphs and to describe how rotation vectors assemble into . The theory yields realizability and convex-structure results for rotation sets, and constructs invariant open sets tied to the class graph, laying the groundwork for a rotational Axiom A-type viewpoint with a companion applications paper for big rotation sets.

Abstract

We develop a rotational hyperbolic theory for surface homeomorphisms. We use the equivalence relation on ergodic measures that have nontrivial rotational behaviour defined in arXiv:2312.06249 to define a rotational counterpart of homoclinic classes. These allows to produce a network of horseshoes representing the whole rotational behaviour f the homeomorphism. We also study the counterpart of heteroclinic connections and give 5 different characterizations of such connections. The main technical tool is the forcing theory of Le Calvez and Tal arXiv:1503.09127, arXiv:1803.04557, and in particular a result of creation of periodic points that can also be seen as a statement of homotopically bounded deviations [GT25a]. This theoretical article is followed by a paper focused of some applications of it to the case of homeomorphisms with big rotation set [Gui25].

Paper Structure

This paper contains 20 sections, 31 theorems, 52 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Forcing theory
  4. Foliations and isotopies.
  5. $\mathcal{F}$-transverse paths and $\mathcal{F}$-transverse intersections.
  6. Recurrence and equivalence.
  7. Accumulation property
  8. Brouwer-Le Calvez foliations and forcing theory
  9. Classification of ergodic rotation sets
  10. Bounded deviations in homotopy
  11. Heteroclinic horseshoes in forcing theory
  12. Markovian intersections
  13. Heteroclinic connections of horseshoes and rotation sets
  14. Creation of heteroclinic connections of horseshoes by forcing theory
  15. Heteroclinic connections between chaotic classes
  16. ...and 5 more sections

Key Result

Theorem 1.4

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 $z \in S$, there exists a geodesic $\gamma_z\subset T^1S$ --- called the tracking geodesic of $z$ ---, and for each

Figures (7)

  • Figure 1: Idea of the statement of Theorem \ref{['ThConnectionHorse']}: if there is a trajectory under the isotopy like the one in the left of the figure in the space of leaves, then there exists two rotational horseshoes for $f$ having a heteroclinic connection.
  • Figure 2: Example of $\widehat{\mathcal{F}}$-transverse intersection.
  • Figure 3: A pre-Markovian intersection (left) and a Markovian intersection(right). The horizontal sub-rectangle for the pre-markovian intersection is denoted $R'_1$.
  • Figure 4: Beginning of the proof of Theorem \ref{['ThConnectionHorse']} for $k_1=3$: construction of the rectangle $\widehat{R}_1$ (top) and Markovian intersections of the image $\widehat{f}^3(\widehat{R}_1)$ with $T_1\widehat{R}_1$, $T_1^2\widehat{R}_1$ and $T_1^3\widehat{R}_1$ (bottom).
  • Figure 5: Proof of Theorem \ref{['ThConnectionHorse']} in the case $k_1 = k_2 = 2$: construction of the rectangles $\widehat{R}_1$ and $\widehat{R}_2$.
  • ...and 2 more figures

Theorems & Definitions (75)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3: Homological rotation sets
  • Theorem 1.4
  • Theorem 1.5: alepablo
  • Definition 1.6
  • Definition 1.7
  • Proposition 1
  • Theorem 2
  • Definition 2.1: $\widehat{\mathcal{F}}$-transverse intersection
  • ...and 65 more