Rigidity of real-analytic actions of $SL(n,\mathbb{Z})$ on $\T^n$: A case of realization of Zimmer program

Abstract

We prove that any real-analytic action of $SL(n,\mathbb{Z}),n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.