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Exponential multiple mixing for commuting automorphisms of a nilmanifold

Part of: Ergodic theory Diophantine approximation, transcendental number theory Locally compact groups and their algebras

Published online by Cambridge University Press:  11 October 2023

TIMOTHÉE BÉNARD  and
PÉTER P. VARJÚ
TIMOTHÉE BÉNARD*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: pv270@dpmms.cam.ac.uk)
PÉTER P. VARJÚ
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: pv270@dpmms.cam.ac.uk)
*
e-mail: tb723@cam.ac.uk
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Abstract

Let $l\in \mathbb {N}_{\ge 1}$ and $\alpha : \mathbb {Z}^l\rightarrow \text {Aut}(\mathscr {N})$ be an action of $\mathbb {Z}^l$ by automorphisms on a compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha (z)$ is ergodic for $z\in \mathbb {Z}^l\smallsetminus \{0\}$ and show that $\alpha $ satisfies exponential n-mixing for any integer $n\geq 2$. This extends the results of Gorodnik and Spatzier [Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), 127–159].

Keywords

exponential mixingmultiple mixingnilmanifold automorphismsSchmidt’s subspace theoremunit equations

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 22D40: Ergodic theory on groups 11J87: Schmidt Subspace Theorem and applications
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 7 , July 2024 , pp. 1729 - 1740
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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