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An almost mixing of all orders property of algebraic dynamical systems

  • L. ARENAS-CARMONA (a1), D. BEREND (a2) and V. BERGELSON (a3)

Abstract

We consider dynamical systems, consisting of $\mathbb{Z}^{2}$ -actions by continuous automorphisms on shift-invariant subgroups of $\mathbb{F}_{p}^{\mathbb{Z}^{2}}$ , where $\mathbb{F}_{p}$ is the field of order $p$ . These systems provide natural generalizations of Ledrappier’s system, which was the first example of a 2-mixing $\mathbb{Z}^{2}$ -action that is not 3-mixing. Extending the results from our previous work on Ledrappier’s example, we show that, under quite mild conditions (namely, 2-mixing and that the subgroup defining the system is a principal Markov subgroup), these systems are almost strongly mixing of every order in the following sense: for each order, one just needs to avoid certain effectively computable logarithmically small sets of times at which there is a substantial deviation from mixing of this order.

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