Exponential mixing of nilmanifold automorphisms

Abstract

We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and regularity of solutions of the cohomological equation for such automorphisms. Our method is based on the quantitative equidistribution results for polynomial maps combined with Diophantine estimates.

Exponential mixing and smooth classification of commuting expanding maps

“…Finally, let us note three more corollaries of exponential mixing, similar to results in [GS14]. We refer there for a more extensive discussion of ideas and background.…”

“…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.…”

“…He also gave counter examples in the C 1category. Gorodnik and Spatzier proved the nilautomorphism version of this result in [GS14].…”

“…Note that they are independent of the p-adic direction. Hence we can use the central limit theorem, Corollary 1.5, and exponential mixing as in [GS14].…”

Higher Order Correlations for Group Actions

“…• to prove the Central Limit Theorem for group actions. Other interesting applications of quantitative bounds on correlations, which we do not discuss here, involve the Kazhdan property (T) [47, §V.4.1], the cohomological equation [50,36,38], the global rigidity of actions [29], and analysis of the distribution of arithmetic counting functions [9,10]. This paper is based on a series of lectures given at the Tata Institue of Fundamental Research which involved participants with quite diverse backgrounds ranging from starting PhD students to senior researchers.…”