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Non-Bernoulli systems with completely positive entropy

Published online by Cambridge University Press:  01 February 2008

A. H. DOOLEY,
V. YA. GOLODETS,
D. J. RUDOLPH and
S. D. SINEL’SHCHIKOV
A. H. DOOLEY
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia (email: a.dooley@unsw.edu.au, golodets@maths.unsw.edu.au)
V. YA. GOLODETS
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia (email: a.dooley@unsw.edu.au, golodets@maths.unsw.edu.au)
D. J. RUDOLPH
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (email: djr@math.umd.edu)
S. D. SINEL’SHCHIKOV
Affiliation:
Institute for Low Temperature Physics & Engineering, Ukrainian National Academy of Sciences, 47 Lenin Avenue, 61103 Kharkov, Ukraine (email: sinelshchikov@ilt.kharkov.ua)
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Abstract

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A new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable group G, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any $h \in (0, \infty ]$, an uncountable family of cpe actions of entropy h, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that if αG is co-induced from an action αΓ of a subgroup Γ, then h(αG)=h(αΓ). We also prove that if αΓ is a non-Bernoulli cpe action of Γ, then αG is also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of $\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008, pp. 87 - 124
Copyright
Copyright © Cambridge University Press 2007

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