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The Teichmüller Space of a 3-Dimensional Anosov Flow

Ruihao Gu, Yi Shi

TL;DR

The paper develops a Teichmüller-type parametrization for smooth orbit-equivalence classes of 3D transitive Anosov flows on closed 3-manifolds by representing orbit data with pairs of Hölder Jacobian data $(f_s,f_u)$ modulo flow-cohomology, thereby realizing the Teichmüller space as a product of function spaces. It proves a Cawley-type realization for flows, establishes path-connectedness of the orbit-equivalence space, and derives rigidity and classification results for flows with $C^1$-smooth strong unstable foliations, including a dichotomy: either constant-roof suspensions or leafwise smoothness of the conjugacy. The key tools are Radon–Nikodym realizations of transverse measures along stable/unstable foliations, Livšic-type cohomology results for cocycles, and careful deformation of HA-flows (time-change and foliation-regularity preserving) to realize prescribed Jacobians. Together, these results provide a concrete, cohomology-based description of the moduli of 3D Anosov flows, with implications for the topology of the orbit-equivalence space and for rigidity classes under smooth foliations.

Abstract

For a $3$-dimensional transitive Anosov flow, we realize its Teichmüller space of smooth orbit-equivalence classes as a product of two Jacobian function spaces. This proves Cawley's theorem [8] for 3-dimensional transitive Anosov flows. As an application, we show the path-connectedness of orbit-equivalence class of $3$-dimensional transitive Anosov flows, which gives a positive answer of Potrie [40, Question 1] in dimension 3. Moreover, we show the stable Jacobian rigidity of time-preserving conjugacy for $3$-dimensional transitive Anosov flows admitting $C^1$-smooth strong unstable foliations.

The Teichmüller Space of a 3-Dimensional Anosov Flow

TL;DR

The paper develops a Teichmüller-type parametrization for smooth orbit-equivalence classes of 3D transitive Anosov flows on closed 3-manifolds by representing orbit data with pairs of Hölder Jacobian data modulo flow-cohomology, thereby realizing the Teichmüller space as a product of function spaces. It proves a Cawley-type realization for flows, establishes path-connectedness of the orbit-equivalence space, and derives rigidity and classification results for flows with -smooth strong unstable foliations, including a dichotomy: either constant-roof suspensions or leafwise smoothness of the conjugacy. The key tools are Radon–Nikodym realizations of transverse measures along stable/unstable foliations, Livšic-type cohomology results for cocycles, and careful deformation of HA-flows (time-change and foliation-regularity preserving) to realize prescribed Jacobians. Together, these results provide a concrete, cohomology-based description of the moduli of 3D Anosov flows, with implications for the topology of the orbit-equivalence space and for rigidity classes under smooth foliations.

Abstract

For a -dimensional transitive Anosov flow, we realize its Teichmüller space of smooth orbit-equivalence classes as a product of two Jacobian function spaces. This proves Cawley's theorem [8] for 3-dimensional transitive Anosov flows. As an application, we show the path-connectedness of orbit-equivalence class of -dimensional transitive Anosov flows, which gives a positive answer of Potrie [40, Question 1] in dimension 3. Moreover, we show the stable Jacobian rigidity of time-preserving conjugacy for -dimensional transitive Anosov flows admitting -smooth strong unstable foliations.
Paper Structure (24 sections, 36 theorems, 189 equations, 4 figures)

This paper contains 24 sections, 36 theorems, 189 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Realization of 3-dimensional Anosov flows
  3. Rigidity of $C^1$ strong hyperbolic foliations
  4. The Teichmüller space of Anosov $3$-flow
  5. Preliminaries
  6. Foliations and flows
  7. Regularity of foliations and flows
  8. Cocycles over flows
  9. Anosov flows
  10. The foliations of Anosov flows
  11. The Livschitz theorem
  12. Equilibrium states of Anosov flows
  13. The Radon-Nikodym realization
  14. Jacobian Flexibility for HA-Flows
  15. The Radon-Nikodym realization for HA-flows
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let ${\cal F}_1$ and ${\cal F}_2$ be transverse foliations of $d$-manifold $M$ with dimension $d_1$ and $d_2$, respectively, where $d_1+d_2=d$. Let ${\cal L}_1$ and ${\cal L}_2$ be also transverse foliations of $M$ with dimension $d_1$ and $d_2$, respectively. Assume that a homeomorphism $H:M\to M$

Figures (4)

  • Figure 1: Commutative Diagram
  • Figure 2: The set $\Sigma_i$, i.e., the piece of the boundary $\partial U_i$ foliated by ${\cal L}$, is divided into rectangles $\Sigma_{ij}$. The curve ${\cal L}^-_{ij}$ is cut by other boundaries of $\Sigma_{ij'}$ into shorter curves.
  • Figure 3: The metric of point $p$ is given by the metric of $p_*^-$ and $p_*^+$.
  • Figure 4: The metric of point $p$ is given by the metric of $p_S$ nad $p_L$.

Theorems & Definitions (104)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1: Journé Lemma J1988
  • Proposition 2.2: H1983PSW1997
  • Proposition 2.3: A2008H1983
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 94 more