Joint integrability and spectral rigidity for Anosov diffeomorphisms

TL;DR

This paper investigates the geometric and spectral properties of Anosov diffeomorphisms on tori, showing that joint integrability of certain subbundles implies conjugacy to linear models and spectral rigidity, with specific results in four dimensions.

Contribution

It establishes conditions under which Anosov diffeomorphisms are conjugate to linear models, revealing spectral rigidity and coherence properties, especially in four-dimensional cases.

Findings

Joint integrability implies conjugacy to linear foliation.

Existence of finest dominated splitting matching linear model.

Spectral rigidity along stable and unstable subbundles.

Abstract

Let $f\colon\mathbb{T}^d\to\mathbb{T}^d$ be an Anosov diffeomorphism whose linearization $A\in{\rm GL}(d,\mathbb{Z})$ is irreducible. Assume that $f$ is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable and unstable subbundles are jointly integrable, then $f$ is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization $A$. Moreover, $f$ admits the finest dominated splitting in weak stable subbundle with dimensions matching those for $A$, and it has spectral rigidity along all these subbundles. In dimension 4 we are also able to obtain a similar result which allows to group the weak stable and unstable subbundles into a center subbundle and assumes joint integrability of strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism $f\in{\rm Diff}^2_{\omega}(\mathbb{T}^4)$ which is $C^1$-close to an irreducible non-conformal automorphism $A\in{\rm Sp}(4,\mathbb{Z})$, the extremal subbundles of $f$ are jointly integrable if and only if $f$ is smoothly conjugate to $A$.