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Mixing and tight polyhedra. (English) Zbl 1167.37304

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 169-175 (2006).
Summary: Actions of \(\mathbb Z^d\) by automorphisms of compact zero-dimensional groups exhibit a range of mixing behaviour. Schmidt introduced the notion of mixing shapes for these systems, and proved that non-mixing shapes can only arise non-trivially for actions on zero-dimensional groups. Masser has shown that the failure of higher-order mixing is always witnessed by non-mixing shapes. Here we show how valuations can be used to understand the (non-)mixing behaviour of a certain family of examples. The sharpest information arises for systems corresponding to tight polyhedra.
For the entire collection see [Zbl 1113.60008].

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
22D40 Ergodic theory on groups
52B11 \(n\)-dimensional polytopes
37A25 Ergodicity, mixing, rates of mixing
28D15 General groups of measure-preserving transformations

Keywords:

mixing; polyhedra
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