Topological centralizers of perturbations of ergodic toral automorphisms with two dimensional center

Abstract

Let $A$ be an ergodic linear automorphism of the torus $\T^N$ whose center space has dimension two. We prove a dichotomy result that for every $f \in \mathcal{U}$, where $\mathcal{U}$ is the $C^{22}$ neighborhood of $A$ inside volume preserving diffeomorphisms on $\T^N$, exactly one of two alternatives holds. Either $f$ has the accessibility property, or $f$ is topologically conjugate to $A$ by a homeomorphism homotopic to the identity which simultaneously conjugates every homeomorphism commuting with $f$ to an affine automorphism of the torus. No irreducibility of the characteristic polynomial of $A$ is assumed. The alternatives exclude each other because a partially hyperbolic diffeomorphism topologically conjugate to $A$ is never accessible, so within $\mathcal{U}$ the failure of accessibility is equivalent to topological conjugacy to $A$. In the second case the centralizer of $f$ in the group of homeomorphisms of the torus is isomorphic to the group of affine transformations commuting with $A$. When the characteristic polynomial of $A$ is irreducible we show that this group is a finite extension of the unit group of an order in the number field generated by an eigenvalue of $A$, hence virtually free abelian of rank $r_1 + r_2 - 1$. The characteristic polynomial is allowed to be reducible, in which case the same description holds with the linear parts ranging over the centralizer of $A$ in $\GL(N,\Z)$. Our method uses an elementary density property of the projection of the lattice to the center space, which replaces irreducibility and which we establish for every ergodic automorphism with no restriction on the dimension of the center space.