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Rigidity of the unstable foliation

Sergi Burniol Clotet

TL;DR

The paper proves a rigidity phenomenon for unstable foliations of transitive Anosov flows on compact 3-manifolds: equivalence of unstable foliations implies topological conjugacy up to a constant time change. The argument splits into two cases via Plante's alternative—constant-time suspensions and topologically mixing flows—and relies on local product structure to force preservation of the center unstable foliation and derive a global time-change $\lambda$. In the suspension case, monodromy conjugacy together with time-scaling yields a conjugacy between the flows up to $t \mapsto \lambda t$; in the mixing case, a detailed construction establishes the same conclusion. This work extends Ratner-type rigidity from horocyclic flows to a broad class of 3D Anosov dynamics, highlighting the rigidity of unstable foliations and the intricate moduli of nonequivalent foliations under perturbations.

Abstract

We establish a rigidity result for the unstable foliations of transitive Anosov flows on 3-manifolds: if the unstable foliations of two such flows are equivalent (that is, if there exists a homeomorphism mapping one foliation to the other), then the flows are topologically conjugate up to a constant change of time. This result partially generalizes earlier rigidity theorems for horocyclic flows on compact surfaces of negative curvature, originating in the work of Ratner. In that setting, it is known that equivalence of unstable foliations implies that the underlying surfaces are homothetic.

Rigidity of the unstable foliation

TL;DR

The paper proves a rigidity phenomenon for unstable foliations of transitive Anosov flows on compact 3-manifolds: equivalence of unstable foliations implies topological conjugacy up to a constant time change. The argument splits into two cases via Plante's alternative—constant-time suspensions and topologically mixing flows—and relies on local product structure to force preservation of the center unstable foliation and derive a global time-change . In the suspension case, monodromy conjugacy together with time-scaling yields a conjugacy between the flows up to ; in the mixing case, a detailed construction establishes the same conclusion. This work extends Ratner-type rigidity from horocyclic flows to a broad class of 3D Anosov dynamics, highlighting the rigidity of unstable foliations and the intricate moduli of nonequivalent foliations under perturbations.

Abstract

We establish a rigidity result for the unstable foliations of transitive Anosov flows on 3-manifolds: if the unstable foliations of two such flows are equivalent (that is, if there exists a homeomorphism mapping one foliation to the other), then the flows are topologically conjugate up to a constant change of time. This result partially generalizes earlier rigidity theorems for horocyclic flows on compact surfaces of negative curvature, originating in the work of Ratner. In that setting, it is known that equivalence of unstable foliations implies that the underlying surfaces are homothetic.

Paper Structure

This paper contains 5 sections, 11 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Proof of the rigidity
  4. Topologically mixing Anosov flows
  5. Constant time suspensions of Anosov diffeomorphisms

Key Result

Theorem 1.1

Marcus83Abe95 Let $(S,g),(S',g')$ be two negatively curved compact surfaces. If the unstable foliations on $T^1S$ and $T^1 S'$ are equivalent, then the two surfaces $(S,g)$ and $(S',g')$ are homothetic.

Figures (2)

  • Figure 1: On the left: the red surface is the image of $\psi$ and the blue surface is the image of $\varphi$. The green segment represents the curve $t\mapsto \varphi(t,0)$. On the right: red surface, blue surface and green segment after applying $g_{-T}$ and restricting the domain.
  • Figure 2: Additivity of $\bar{t}_2$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Local product structure
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 7 more