Rigidity of center Lyapunov exponents and $su$-integrability

TL;DR

This paper proves that for certain conservative partially hyperbolic diffeomorphisms on the 3-torus, joint integrability of stable and unstable bundles is characterized by uniform center Lyapunov exponents, confirming a conjecture.

Contribution

It establishes a rigidity result linking bundle integrability to Lyapunov exponents and confirms the Ergodic Conjecture for these systems.

Findings

Stable and unstable bundles are jointly integrable iff all periodic points have the same center Lyapunov exponent.

Such diffeomorphisms are ergodic.

The Ergodic Conjecture on $ ext{T}^3$ is proved.

Abstract

Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on $\mathbb{T}^3$, is ergodic. This proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on $\mathbb{T}^3$.