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Rigidity of real-analytic actions of \(SL(n,\mathbb Z)\) on \(\mathbb T^n\): a case of realization of Zimmer program. (English) Zbl 1192.37025

The group \(SL(n,\mathbb Z)\) acts on the \(n\)-dimensional torus by automorphisms which are projections of the linear maps in \({\mathbb R}^n\) (standard action). The corresponding action on \({\mathbb Z}^n\) is the standard homotopy data. We prove that any real-analytic action of \(SL(n,\mathbb Z),n\geq 3\) 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.

MSC:

37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
22F05 General theory of group and pseudogroup actions
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