# An almost mixing of all orders property of algebraic dynamical systems Academic Article

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### 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.

### publication date

• September 7, 2017