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Smooth rigidity for 3-dimensional dissipative Anosov flows

Andrey Gogolev, Martin Leguil, Federico Rodriguez Hertz

TL;DR

This work analyzes rigidity of conjugacies between dissipative 3D Anosov flows under $C^0$-conjugacy. The authors develop a technical framework based on period asymptotics, hitting-time jets, adapted charts, and stable/unstable templates to relate period data to eigendata, leading to a dichotomy: either the conjugacy is smoothly conjugate or it swaps the SRB measures; a third possibility involves partial regularity of foliations. They establish generic and local rigidity results, sharpen Jacobian rigidity results for flows and diffeomorphisms, and prove Teichmüller-space stratification phenomena on the 2-torus, including non-surjectivity of certain regularity maps via Cawley-type realizations. The paper also furnishes detailed examples illustrating SRB-swapping and non-C^1 conjugacies, and outlines numerous open questions about exceptional dynamics and higher-regularity rigidity. Overall, the results deepen our understanding of how period and Jacobian data govern smooth conjugacy and the statistical structure of 3D Anosov flows, with consequences for the geometry of 3-manifolds and the structure of associated Teichmüller-type spaces.

Abstract

We consider two transitive $3$-dimensional Anosov flows which do not preserve volume and which are continuously conjugate to each other. Then, disregarding certain exceptional cases, such as flows with $C^1$ regular stable or unstable distributions, we prove that either the conjugacy is smooth or it sends the positive SRB measure of the first flow to the negative SRB measure of the second flow and vice versa. We give a number of corollaries of this result. In particular, we establish local rigidity on a $C^1$-open $C^\infty$-dense subspace of transitive Anosov flows; we improve the classical de la Llave-Marco-Moriyón rigidity theorem for dissipative Anosov diffeomorphisms on the $2$-torus by merely assuming matching of (full) Jacobian data at periodic points; we also exhibit the first evidence that the Teichmüller space of smooth conjugacy classes of Anosov diffeomorphisms on the $2$-torus is well-stratified according to regularity.

Smooth rigidity for 3-dimensional dissipative Anosov flows

TL;DR

This work analyzes rigidity of conjugacies between dissipative 3D Anosov flows under -conjugacy. The authors develop a technical framework based on period asymptotics, hitting-time jets, adapted charts, and stable/unstable templates to relate period data to eigendata, leading to a dichotomy: either the conjugacy is smoothly conjugate or it swaps the SRB measures; a third possibility involves partial regularity of foliations. They establish generic and local rigidity results, sharpen Jacobian rigidity results for flows and diffeomorphisms, and prove Teichmüller-space stratification phenomena on the 2-torus, including non-surjectivity of certain regularity maps via Cawley-type realizations. The paper also furnishes detailed examples illustrating SRB-swapping and non-C^1 conjugacies, and outlines numerous open questions about exceptional dynamics and higher-regularity rigidity. Overall, the results deepen our understanding of how period and Jacobian data govern smooth conjugacy and the statistical structure of 3D Anosov flows, with consequences for the geometry of 3-manifolds and the structure of associated Teichmüller-type spaces.

Abstract

We consider two transitive -dimensional Anosov flows which do not preserve volume and which are continuously conjugate to each other. Then, disregarding certain exceptional cases, such as flows with regular stable or unstable distributions, we prove that either the conjugacy is smooth or it sends the positive SRB measure of the first flow to the negative SRB measure of the second flow and vice versa. We give a number of corollaries of this result. In particular, we establish local rigidity on a -open -dense subspace of transitive Anosov flows; we improve the classical de la Llave-Marco-Moriyón rigidity theorem for dissipative Anosov diffeomorphisms on the -torus by merely assuming matching of (full) Jacobian data at periodic points; we also exhibit the first evidence that the Teichmüller space of smooth conjugacy classes of Anosov diffeomorphisms on the -torus is well-stratified according to regularity.

Paper Structure

This paper contains 42 sections, 58 theorems, 333 equations, 3 figures.

Table of Contents

  1. Overview
  2. Detailed statements of results
  3. Rigidity problem for Anosov flows and prior results
  4. Main technical result
  5. Generic rigidity, local rigidity and rigidity from matching of abstract potentials
  6. Stratification of the Teichmüller space of smooth conjugacy classes according to regularity
  7. Further questions
  8. Structure of the paper
  9. Background knowledge on adapted charts and templates
  10. Notation
  11. Adapted charts
  12. Hitting times and their jets
  13. Templates and regularity of $E^s$ and $E^u$
  14. Asymptotic formula for periods
  15. Shadowing setup
  16. ...and 27 more sections

Key Result

Theorem 1

Let $X^t\colon M\to M$ and $Y^t\colon N\to N$ be two transitive dissipative $C^\infty$ Anosov flows on $3$-manifolds $M$ and $N$. Assume that they are $C^0$-conjugate by a homeomorphism $\Phi\colon M \to N$, $\Phi \circ X^t=Y^t \circ \Phi$. Assume that for any periodic point $p=X^T(p)$ Jacobians mat Then, $X^t$ and $Y^t$ are $C^{\infty}$-conjugate.

Figures (3)

  • Figure 1: Transveral $\Sigma_x$.
  • Figure 2: The template can be informally thought of as a function measuring the angle between the local stable manifold and the transversal $\Sigma_x$.
  • Figure 3: The construction of quasi-Fuchsian flows by "taking the bracket" of the flows $X^t$ and $\bar{X}^t$ to produce conjugate flows $Y^t$ and $Z^t$.

Theorems & Definitions (150)

  • Theorem 1
  • Corollary 2
  • Remark 1.1
  • Theorem 3
  • Corollary 4
  • Theorem 2.1: De la Llave-Moriyón dllMordllSRB, Pollicott pollicott1988
  • Definition 2.2: $k$-pinching
  • Definition 2.3
  • Remark 2.4
  • Theorem 5
  • ...and 140 more