Topological invariance of Liouville structures for taut foliations and Anosov flows

TL;DR

The paper develops a topological-invariance theory for Liouville structures arising from taut, hypertaut foliations on closed 3-manifolds, showing that the Liouville thickening on $V=[-1,1]\times M$ is invariant (up to Liouville deformation) under topological conjugacy of foliations. It specializes to weak foliations of Anosov flows, proving invariance under orbit equivalence and, via bicontact structures and projectively Anosov deformations, deformation equivalence of the associated dynamical data. The technical core combines a smoothing scheme for $C^1$ foliations and a refined Vogel-type uniqueness result for contact approximations, together with a deformation argument to convert pre-Liouville structures into genuine Liouville thickenings. These results bridge foliation theory, contact/symplectic geometry, and 3-manifold dynamics, yielding new invariant structures and a pathway toward classifying transitive partially hyperbolic diffeomorphisms in dimension three. The appendix extends smoothing methods to construct new collapsed Anosov-flow examples, contributing to the broader program of understanding priors and deformations in 3D dynamics.

Abstract

Building on the work of Eliashberg and Thurston, we associate to a taut foliation on a closed oriented $3$-manifold $M$ a Liouville structure on the thickening $[-1,1] \times M$, under suitable hypotheses. Our main result shows that this Liouville structure is a topological invariant of the foliation: two such foliations which are topologically conjugated induce exact symplectomorphic Liouville structures. Specializing to the case of weak foliations of Anosov flows, we obtain that under natural orientability conditions, the Liouville structures originally introduced by Mitsumatsu are invariant under orbit equivalence. Our methods also imply that two orbit equivalent Anosov flows are deformation equivalent through projectively Anosov flows. The proofs combine two main technical ingredients: (1) a careful smoothing scheme for topological conjugacies between $C^1$-foliations, and (2) a refinement of a deep result of Vogel on the uniqueness of contact structures approximating a foliation. In an appendix, this smoothing scheme is used to construct new examples of collapsed Anosov flows, providing a key step to complete the classification of transitive partially hyperbolic diffeomorphisms in dimension three.

Figures (8)

• Figure 1: Summary of Vogel's theorem.
• Figure 2: Local depiction of $h$.
• Figure 3: Neighborhoods of simplices.
• Figure 4: Steps of the smoothing procedure.
• Figure 5: Blueprint of the setup of Vogel's proof.

Theorems & Definitions (104)

• Definition 1
• Definition 3
• Example 4
• Theorem A: $C^0$-functoriality
• Theorem B
• Theorem 5
• Definition 6
• Theorem 7
• Corollary 8: Anosov bifoliations
• Theorem : Vogel V16