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

TL;DR

This paper proves that for certain Anosov diffeomorphisms on the 3-torus close to a hyperbolic automorphism with complex eigenvalues, a conjugacy matching periodic data is necessarily smooth.

Contribution

It establishes smoothness of conjugacies near specific hyperbolic automorphisms on $ ext{T}^3$ when periodic data match.

Findings

Matching periodic data implies $C^{1+\text{H"older}}$ conjugacy near the automorphism.

Smoothness result applies to diffeomorphisms close to hyperbolic automorphisms with complex eigenvalues.

The result extends rigidity phenomena in Anosov systems to a neighborhood of certain linear automorphisms.

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 $C^{1+\text{H\"older}}$