Mixing properties of commuting nilmanifold automorphisms

Abstract

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every nontrivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.

Exponential multiple mixing for commuting automorphisms of a nilmanifold

“…Theorem 1.1 for mixing without a rate, that is the statement that ergodicity of α(z) for every z ∈ Z l {0} implies that α is n-mixing for all n ≥ 2, was known before. It is due to Parry (1969 [16]) for l = 1, K. Schmidt-Ward (1993 [18]) in the case where N is a torus and l ≥ 2, and Gorodnik-Spatzier (2015 [10]) in general.…”

Exponential mixing and smooth classification of commuting expanding maps

“…In the case of Z k actions by ergodic automorphisms on nilmanifolds, exponential mixing was established by Gorodnik and Spatzier [GS14] and [GS15], based on the work of Green and Tao [GT12,GT14]. In our case, the structure of S(M) is essentially different from a real nilmanifold.…”

Exponential mixing of nilmanifold automorphisms

“…• Our techniques also allow to establish exponential mixing properties for Z k -actions by automorphisms of nilmanifolds when k ≥ 2. Since this requires more delicate Diophantine estimates, we pursue this in a sequel paper [11]. This result has found a striking application to the problem of global rigidity of smooth actions.…”