Centralizer classification and rigidity for some partially hyperbolic toral automorphisms

In this paper we consider local centralizer classification and rigidity of some toral automorphisms. In low dimensions we classify up to finite index possible centralizers for volume preserving diffeomorphisms $f$ $C^1-$close to an ergodic irreducible toral automorphism $L$. Moreover, we show a rigidity result in the case that the centralizer of $f$ is large: If the smooth centralizer $Z^{\mathrm{\infty }}(f)$ is virtually isomorphic to that of $L$ then $f$ is $C^{\mathrm{\infty }}-$conjugate to $L$. In higher dimensions we show a similar rigidity result for certain irreducible toral automorphisms. We also classify up to finite index all possible centralizers for symplectic diffeomorphisms $C^5-$close to a class of irreducible symplectic automorphisms on tori of any dimension.