## Acta Mathematica

### Mixing properties of commuting nilmanifold automorphisms

#### Abstract

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial 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.

#### Note

A. G. was supported in part by EPSRC grant EP/H000091/1 and ERC grant 239606. R. S. was supported in part by NSF grant DMS-0906085.

#### Article information

Source
Acta Math., Volume 215, Number 1 (2015), 127-159.

Dates
Revised: 11 July 2014
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.acta/1485802444

Digital Object Identifier
doi:10.1007/s11511-015-0130-0

Mathematical Reviews number (MathSciNet)
MR3413978

Zentralblatt MATH identifier
1360.37010

Rights

#### Citation

Gorodnik, Alexander; Spatzier, Ralf. Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), no. 1, 127--159. doi:10.1007/s11511-015-0130-0. https://projecteuclid.org/euclid.acta/1485802444

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