Local smooth rigidity of Anosov diffeomorphisms in $\mathbb{T}^{3}$

Abstract: Given a $C^0$ conjugacy between two Anosov diffeomorphisms, the matching periodic data problem asks whether this conjugacy is smooth provided spectral data of the diffeomorphisms match at periodic points. We show that if the two $C^0$ conjugate diffeomorphisms on $\mathbb{T}^3$ are sufficiently close to a hyperbolic linear automorphism with a pair of complex conjugate eigenvalues, then the conjugacy must be smooth. In particular, we have that in a neighborhood of a hyperbolic toral automorphism, matching periodic data implies that the conjugacy is $C^{1+\text{Hölder}}$

• The paper establishes local smooth rigidity by proving that matching periodic data guarantees smooth conjugacy near a hyperbolic automorphism with complex eigenvalues.
• It leverages holonomy and cohomology techniques, including quadrilateral holonomy and Parry monoid representations, to handle the non-linear case.
• The study applies bootstrapping via Journé’s lemma and an entropy condition (SRB = MME) to ensure the required regularity of the conjugacy.

Local Smooth Rigidity of Anosov Diffeomorphisms on $\mathbb{T}^3$

Overview

The paper "Local smooth rigidity of Anosov diffeomorphisms in $\mathbb{T}^3$" (2603.29155) addresses the longstanding question of periodic data rigidity for Anosov diffeomorphisms on the 3-torus. Specifically, the work rigorously establishes local smooth rigidity for pairs of $C^r$ Anosov diffeomorphisms that are $C^0$-conjugate and have matching periodic data in a neighborhood of a hyperbolic automorphism with a pair of complex conjugate eigenvalues. It advances the theory by bridging the previously unresolved case when both diffeomorphisms are non-linear and the periodic data are non-constant, provided the SRB (Sinai-Ruelle-Bowen) measure coincides with the measure of maximal entropy.

Background and Problem Formulation

Anosov diffeomorphisms on compact manifolds embody the classical archetype of chaotic dynamical systems due to their structural stability and rich ergodic properties. Of specific interest are Anosov diffeomorphisms on $\mathbb{T}^3$ arising as small perturbations of hyperbolic toral automorphisms. The local smooth rigidity or matching periodic data problem is fundamental in differential rigidity theory: is the topological conjugacy between two such diffeomorphisms necessarily smooth if the local dynamical behavior (Lyapunov spectra at periodic points) is matched?

A necessary condition for $C^1$-regularity of the conjugacy is matching periodic data, that is, for every periodic point, the derivatives of the two maps are conjugate. The sufficiency of this condition, especially in higher dimensions and non-algebraic settings, has remained open except for certain cases. The primary technical challenge arises when the unstable (or stable) bundle becomes multidimensional, and the spectrum contains complex conjugate eigenvalues.

Main Results

The principal contributions are articulated in the following theorems:

1. Local Smooth Rigidity for Non-Constant Matching Periodic Data:

If $f$, $g$ are $C^r$ Anosov diffeomorphisms on $\mathbb{T}^3$, $\mathbb{T}^3$0-close to a hyperbolic automorphism $\mathbb{T}^3$1 with a pair of complex conjugate eigenvalues, and have non-constant matching periodic data, and if $\mathbb{T}^3$2 has SRB measure equal to the measure of maximal entropy, then the topological conjugacy $\mathbb{T}^3$3 is $\mathbb{T}^3$4 smooth (with $\mathbb{T}^3$5 if $\mathbb{T}^3$6, $\mathbb{T}^3$7 otherwise).

2. General Local Rigidity:

In the same setting, any pair of $\mathbb{T}^3$8 Anosov diffeomorphisms on $\mathbb{T}^3$9, $C^r$0-close to $C^r$1, that have matching periodic data, are $C^r$2-conjugate.

A crucial aspect is the removal of any requirement that either $C^r$3 or $C^r$4 be linear or have constant periodic data, thereby closing the main gap in the literature for dimension three in the presence of complex eigenvalues. The regularity statement aligns with classical bootstrapping methods and the regularity of dynamical foliations.

Bold Claim: In a $C^r$5-neighborhood of a hyperbolic automorphism with complex eigenvalues, matching periodic data is sufficient to guarantee smooth conjugacy between $C^r$6 Anosov diffeomorphisms, under the entropy assumption.

Technical Contributions

Central to the argument are the following innovations and technical apparatus:

1. Holonomy and Cohomology Techniques: The authors leverage the fiber-bunching property of the unstable derivative cocycle, ensuring the well-definedness of Hölder continuous holonomy maps along stable and unstable foliations. Regularity properties (up to $C^r$7 or $C^r$8) are bootstrapped using normal form theory and control of cocycle dynamics in the fiber.
2. Quadrilateral HolonomY: The construction of a quadrilateral holonomy map allows for encoding fine regularity properties of the conjugacy by tracking holonomies along special quadrilaterals in the $C^r$9-foliations. Proving that these maps and their conjugates share smooth structures is pivotal for amplifying local differentiability of the conjugacy.
3. Parry Monoids and Trace Rigidity: The approach introduces the Parry monoid, generated by homoclinic $C^0$0-loops, and develops a representation-theoretic technique to compare cocycles over periodic orbits, reducing smooth rigidity to a character-theoretic problem for holonomy representations. A dichotomy is established: either one cocycle is an (almost) coboundary or the holonomy representations are isomorphic, in which case cocycle conjugacy (and thus conjugacy of the dynamics) follows.
4. Bootstrapping via Journé's Lemma: The authors show that local regularity along foliations (stable and unstable) can be promoted to overall smoothness of the conjugacy using a variant of Journé’s lemma, provided necessary Hölder and absolute continuity regularity is established.
5. Entropy Condition: The SRB = MME condition ensures that the relevant cocycle has cohomologically trivial determinant, an essential algebraic input for the cocycle classification and normalization to the special linear group.

Numerical and Conceptual Implications

While the paper is primarily qualitative, the structure of the quadrilateral holonomy method allows one to extract explicit bounds on the neighborhoods of linear models and provides effective control over the regularity loss in coboundary and bootstrapping arguments. The dichotomy result yields a sharp criterion: Either all nontrivial holonomic data are smoothed out by conjugacy, or the dynamical system is algebraically trivial up to coboundary.

The results unify and strengthen previous local rigidity theorems by de la Llave, Kalinin-Sadovskaya, and Gogolev et al. The approach also emphasizes the crucial distinction between the cases of real and complex eigenvalues, with the complex case being uniquely amenable to these rigidity arguments due to irreducibility properties of rotation subgroups in $C^0$1.

Broader Context and Open Questions

The implications are primarily geometric and algebraic: the theorem establishes that local dynamical data (periodic derivative spectra) capture global smooth conjugacy type in the open set of Anosov diffeomorphisms $C^0$2-close to a given linear model with complex spectrum. This is particularly significant given known counterexamples in dimension four and above.

The work leaves open the periodic data rigidity problem for higher dimensions ($C^0$3) and for real spectrum cases in dimension three with nontrivial periodic data, referencing previous work that indicates substantial obstructions in those settings.

A key open question remains: Are all Anosov diffeomorphisms on $C^0$4 $C^0$5 periodic data rigid? Positive resolution would have dramatic consequences for rigidity theory and the classification of higher rank abelian group actions.

Conclusions

This paper rigorously resolves the local smooth rigidity conjecture for $C^0$6 Anosov diffeomorphisms on $C^0$7 near a linear model with complex conjugate eigenvalues, under the necessary entropy and periodic data assumptions (2603.29155). The quadrilateral holonomy approach and the Parry representation technique develop a robust framework that could inform further progress in higher dimensions or more general dynamical settings. The results substantively advance the understanding of smooth classification problems in hyperbolic dynamics, marking a definitive step in the periodic data rigidity program for three-dimensional tori.