[Paper Review] Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity

Andrey Gogolev, Federico Rodriguez Hertz|arXiv (Cornell University)|Oct 5, 2022

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper establishes a new smooth rigidity theorem for 3-dimensional volume-preserving Anosov flows: if two such flows are topologically conjugate, the conjugacy is smooth unless both are constant roof suspensions of Anosov torus automorphisms. The result removes the need to assume matching stable/unstable eigenvalues at periodic orbits, relying instead solely on period matching, with applications to marked length spectrum rigidity and Anosov diffeomorphism rigidity on the 2-torus.

ABSTRACT

Let $X_1^t$ and $X_2^t$ be volume preserving Anosov flows on a 3-dimensional manifold $M$. We prove that if $X_1^t$ and $X_2^t$ are $C^0$ conjugate then the conjugacy is, in fact, smooth, unless $M$ is a mapping torus of an Anosov automorphism of $\mathbb T^2$ and both flows are constant roof suspension flows. We deduce several applications. Among them is a new result on rigidity of Anosov diffeorphisms on $\mathbb T^2$ and a new "weighted" marked length spectrum rigidity result for surfaces of negative curvature.

Motivation & Objective

• To establish smooth rigidity for 3-dimensional volume-preserving Anosov flows under weaker assumptions than previously known.
• To determine whether period matching alone (without eigenvalue matching) implies smooth conjugacy.
• To resolve the structural rigidity problem in low-dimensional hyperbolic dynamics by identifying the only obstruction to smooth conjugacy.
• To apply the result to prove new rigidity theorems for Anosov diffeomorphisms on the 2-torus and for weighted marked length spectra on negatively curved surfaces.

Proposed method

• Use of a conjugacy between two $ C^r $, $ r>2 $, volume-preserving Anosov flows on a 3-manifold as the starting point.
• Analysis of the obstructions to smoothness of the conjugacy, focusing on eigenvalues of the linearized Poincaré return maps at periodic orbits.
• Application of Journé's regularity lemma to upgrade $ C^0 $ conjugacy to $ C^{r_*} $ smoothness under generic conditions.
• Identification of the exceptional case: when both flows are constant roof suspensions of Anosov automorphisms on $ \mathbb{T}^2 $, where eigenvalue matching becomes necessary.
• Use of equilibrium states and thermodynamic formalism to analyze the weighted marked length spectrum, leveraging the cohomological properties of the cocycle difference.
• Application of the Alternate Livshits Theorem to conclude that a cocycle difference being cohomologous to zero implies the original cocycle is a coboundary.

Experimental results

Research questions

• RQ1 Under what conditions does a $ C^0 $ conjugacy between two 3-dimensional volume-preserving Anosov flows imply smooth conjugacy?
• RQ2 Can the smoothness of the conjugacy be established without assuming matching stable and unstable eigenvalues at corresponding periodic orbits?
• RQ3 What is the precise obstruction to smooth conjugacy in the absence of eigenvalue matching, and is it minimal?
• RQ4 Can the result be extended to rigidity of the weighted marked length spectrum on negatively curved surfaces?
• RQ5 Does the conjugacy between $ C^r $ Anosov flows on $ \mathbb{T}^2 $ imply smoothness without eigenvalue matching, and what does this imply for the dynamics?

Key findings

• A $ C^0 $ conjugacy between two $ C^r $, $ r>2 $, volume-preserving Anosov flows on a 3-manifold is $ C^{r_*} $ smooth unless both flows are constant roof suspensions of Anosov automorphisms on $ \mathbb{T}^2 $.
• The only obstruction to smooth conjugacy is the exceptional case of constant roof suspensions, where eigenvalue matching is necessary.
• The result implies a new rigidity theorem for $ C^r $ Anosov diffeomorphisms on $ \mathbb{T}^2 $: conjugacy implies smooth conjugacy.
• A new weighted marked length spectrum rigidity result is established for negatively curved surfaces, where the spectrum weighted by a Hölder cocycle determines the metric up to isometry.
• The proof relies on the fact that if equilibrium states for two cocycles are equal, then their difference is cohomologous to zero, which implies the cocycle is a coboundary via the Alternate Livshits Theorem.
• The key technical step uses a pigeonhole principle argument to show that a certain map from $ \{1,\dots,N+1\}^N $ to a finite set of measures must have two distinct preimages mapping to the same measure, enabling the cohomological conclusion.