Spectrum rigidity and integrability for Anosov diffeomorphisms

Let $f$ be a partially hyperbolic derived-from-Anosov diffeomorphism on 3-torus $\mathbb{T}^3$. We show that the stable and unstable bundle of $f$ is jointly integrable if and only if $f$ is Anosov and admits spectrum rigidity in the center bundle. This proves the Ergodic Conjecture on $\mathbb{T}^3$.
In higher dimensions, let $A\in{\rm SL}(n,\mathbb{Z})$ be an irreducible hyperbolic matrix admitting complex simple spectrum with different moduli, then $A$ induces a diffeomorphism on $\mathbb{T}^n$. We will also discuss the equivalence of integrability and spectrum rigidity for $f\in{\rm Diff}^2(\mathbb{T}^n)$ which is $C^1$-close to $A$.