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A group from a map and orbit equivalence

Jérôme Los, Natalia A. Viana Bedoya

TL;DR

The paper investigates when an expanding piecewise circle map $Φ$ can arise as a Bowen–Series-type map for a discrete group acting on $S^1$. By imposing dynamical conditions EC, (E+), (E−) and CS-$\lambda$ (with $\lambda>1$), it builds a generating set from $Φ$, proves the associated group $G_{X_{Φ}}$ is Gromov-hyperbolic with boundary $S^1$, and shows the group is a torsion-free surface group acting geometrically on a hyperbolic-like space. It also proves an orbit equivalence between $Φ$ and $G_{X_{Φ}}$, and provides a direct appendix showing the group is conjugate to a Fuchsian surface group, thereby connecting the dynamics to classical hyperbolic geometry and entropy via the algebraic number $\lambda$. The results establish a partial converse to Bowen–Series constructions and reveal a rigid, piecewise-affine structure for the resulting surface groups, linking topological entropy to volume entropy of a geometric presentation. Overall, the work opens a dynamical-spaces framework for constructing and understanding surface groups from circle dynamics and clarifies entropy-growth relationships in this setting.

Abstract

In two papers published in 1979, R. Bowen and C. Series defined a dynamical system from a Fuchsian group, acting on the hyperbolic plane $\mathbb{H}^2$. The dynamics is given by a map on $S^1$ which is, in particular, an expanding piecewise homeomorphism of the circle. In this paper we consider a reverse question: which dynamical conditions for an expanding piecewise homeomorphism of $S^1$ are sufficient for the map to be a ``Bowen-Series-type" map (see below) for some group $G$ and which groups can occur? We give a partial answer to these questions.

A group from a map and orbit equivalence

TL;DR

The paper investigates when an expanding piecewise circle map can arise as a Bowen–Series-type map for a discrete group acting on . By imposing dynamical conditions EC, (E+), (E−) and CS- (with ), it builds a generating set from , proves the associated group is Gromov-hyperbolic with boundary , and shows the group is a torsion-free surface group acting geometrically on a hyperbolic-like space. It also proves an orbit equivalence between and , and provides a direct appendix showing the group is conjugate to a Fuchsian surface group, thereby connecting the dynamics to classical hyperbolic geometry and entropy via the algebraic number . The results establish a partial converse to Bowen–Series constructions and reveal a rigid, piecewise-affine structure for the resulting surface groups, linking topological entropy to volume entropy of a geometric presentation. Overall, the work opens a dynamical-spaces framework for constructing and understanding surface groups from circle dynamics and clarifies entropy-growth relationships in this setting.

Abstract

In two papers published in 1979, R. Bowen and C. Series defined a dynamical system from a Fuchsian group, acting on the hyperbolic plane . The dynamics is given by a map on which is, in particular, an expanding piecewise homeomorphism of the circle. In this paper we consider a reverse question: which dynamical conditions for an expanding piecewise homeomorphism of are sufficient for the map to be a ``Bowen-Series-type" map (see below) for some group and which groups can occur? We give a partial answer to these questions.
Paper Structure (21 sections, 45 theorems, 40 equations, 10 figures)

This paper contains 21 sections, 45 theorems, 40 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A class of piecewise homeomorphisms on $S^1$
  3. The class of maps
  4. Elementary properties of the permutations $\delta$ and $\gamma$
  5. Construction of a group from the map $\Phi$
  6. A toy model construction of diffeomorphisms from $\Phi$
  7. Dynamical properties of $\Phi$
  8. Affine extensions
  9. A parametrised extension family from $\Phi$
  10. Generators and relations from $\Phi$
  11. Some metric spaces associated to $\Phi$
  12. A first space: $\Gamma^0_{\Phi}$
  13. The dynamical graph $\Gamma_{\Phi}$
  14. An action of $G_{X_{\Phi}}$ on $\Gamma_{\Phi}$
  15. A preliminary step
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\Phi: S^1 \rightarrow S^1$ be an orientation preserving discontinuous piecewise homeomorphism satisfying the conditions: $\rm (EC), (E+), (E-), (CS{\textrm{-}}\lambda)$, for some $\lambda>1$. Then there exists a discrete subgroup $G_{\Phi}$ of ${\rm Homeo}^+ (S^1)$ such that:

Figures (10)

  • Figure 1: Condition (SE)
  • Figure 2: The connect-the-dots construction of $f_{j} \in {\rm Diff}^+ (S^1)$
  • Figure 3: The neighborhood $\widetilde{V}_j$ and its image by $\widetilde{\Phi}^{k(j)}$
  • Figure 4: Variation interval equalities for $k(j) =3$, with $\iota(j) = \overline{j}$ and $\zeta^{\pm 1}(j) = j \pm 1$
  • Figure 5: The partition (\ref{['PartitionAD']}) in the proof of Lemma \ref{['PartitionVarphi']} for $k(j) =3$.
  • ...and 5 more figures

Theorems & Definitions (98)

  • Theorem
  • Theorem
  • Lemma 2.1
  • Remark 1
  • Lemma 2.2
  • proof
  • Remark 2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 88 more