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 ∈ (0, ∞], 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 α G is co-induced from an action α of a subgroup , then h(α G ) = h(α ). We also prove that if α is a non-Bernoulli cpe action of , then α G 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 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.Another approach to constructing non-Bernoulli K -automorphisms was due to Feldman [7], who introduced the concept of a loosely Bernoulli system. A loosely Bernoulli action of positive entropy is one that is Kakutani equivalent to a Bernoulli shift. Feldman demonstrated the existence of K -automorphisms which are not loosely Bernoulli. This area was further investigated by Ornstein et al [25] who, in particular, produced an uncountable family of K -automorphisms which pairwise are not Kakutani equivalent. We mention also contributions of Katok to this program [19,20]. Perhaps the simplest example of a K -automorphism S, which is not loosely Bernoulli, was given by Kalikow [16] in his famous study of T, T −1 actions. Kalikow's example has the property that S is isomorphic to S −1 .Recently, Hoffman [15] developed a new and systematic approach to the problem of producing non-Bernoulli K -automorphisms, and many further properties of non-Bernoulli automorphisms have been demonstrated in the literature [15,35,36].As we shall outline below, the theory of entropy and of cpe actions of amenable groups is now rather well developed. It is thus natural to ask whether an infinite amenable group must have non-Bernoulli cpe actions. In this article, we shall extend the theorems of Ornstein and Shields [26] to actions of amenable groups which have an element of infinite order.The question remains open for infinite amenable groups, all of whose elements are of finite order.A constructive approach to the entropy of actions of locally-compact amenable groups is due to Ornstein and Weiss [27], Weiss [40,41] and Lindenstrauss and Weiss [21]. They developed the theory of tiles and quasi-tiles in amenable groups, which allowed them to generalize some key results ...

We prove that an ergodic free action of a countable discrete amenable group with completely positive entropy has a countable Lebesgue spectrum. Our approach is based on the Rudolph-Weiss result on the equality of conditional entropies for actions of countable amenable groups with the same orbits. Relative completely positive entropy actions are also considered. An application to the entropic properties of Gaussian actions of countable discrete abelian groups is given. # 2002 Elsevier Science (USA)

Fundamental groups of ergodic dynamical systems with invariant measure are considered.

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The entropy for actions of a finitely-generated nilpotent group G is investigated. The Pinsker algebra of such actions is described explicitly. The systems with completely positive entropy are shown to have a sort of 'asymptotic independence property', just as in the case of Z d-actions. The invariant partitions are used to prove that the property of completely positive entropy is equivalent to the property of the K-system (the property of the existence of special 'good' partitions). A complete spectral characterization of K-systems is given. A construction of examples of completely positive non-Bernoullian actions of general countable nilpotent groups and a class of solvable groups is presented.