TY - JOUR

T1 - Non-Bernoulli systems with completely positive entropy

AU - Dooley, Anthony H.

AU - Golodets, V. Ya.

AU - Rudolph, D. J.

AU - Sinel’shchikov, S. D.

PY - 2008/2/1

Y1 - 2008/2/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38149050734&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1017/S014338570700034X

U2 - 10.1017/S014338570700034X

DO - 10.1017/S014338570700034X

M3 - Article

VL - 28

SP - 87

EP - 124

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 01

ER -