Bootstrap for local rigidity of Anosov automorphisms on the 3-torus

TL;DR

This paper proves a strong local rigidity result for hyperbolic automorphisms of the 3-torus with real spectrum, showing that small perturbations are smoothly conjugate under certain eigenvalue conditions.

Contribution

It extends 2D local rigidity theory to 3D, establishing conditions for smooth conjugacy of perturbed automorphisms with real spectrum.

Findings

Perturbations of hyperbolic automorphisms are smoothly conjugate if eigenvalue obstructions vanish.

Completes the local rigidity program for 3D hyperbolic automorphisms.

Combines with prior results to extend rigidity theory to higher dimensions.

Abstract

We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let $L\colon\mathbb T^3\to\mathbb T^3$ be a hyperbolic automorphism of the 3-torus with real spectrum and let $f$ be a $C^1$ small perturbation of $L$. Then $f$ is smoothly ($C^\infty$) conjugate to $L$ if and only if obstructions to $C^1$ conjugacy given by the eigenvalues at periodic points of $f$ vanish. By combining our result and a local rigidity result of Kalinin and Sadovskaya for conformal automorphisms this completes the local rigidity program for hyperbolic automorphisms in dimension 3. Our work extends de la Llave-Marco-Moriy\'on 2-dimensional local rigidity theory.