On the entropy theory of finitely-generated nilpotent group actions

Golodets, V. Ya.; Sinel'shchikov, S.

doi:10.1017/s0143385702001104

Cited by 8 publications

(12 citation statements)

References 11 publications

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“…In this section, we briefly consider the entropy of actions of nilpotent countable torsion-free groups with a finite number of generators [17]. In this section, we briefly consider the entropy of actions of nilpotent countable torsion-free groups with a finite number of generators [17].…”

Section: 1mentioning

confidence: 99%

“…Now, (F n ), n ∈ N is a Følner sequence of sets in G [17]. Now, (F n ), n ∈ N is a Følner sequence of sets in G [17].…”

Section: 1mentioning

confidence: 99%

“…Dooley and Golodets [11] proved that any CPE action of a countable amenable group has Lebesgue spectrum with infinite multiplicity. Dooley et al [12] studied non-Bernoulli CPE actions of countable amenable groups constructed as coinduced actions: a version of this construction also appeared in [17] in connection with a question of Thouvenot on CPE actions for discrete nilpotent groups. Glasner et al [16] studied Pinsker algebras in this setting: Danilenko and Park [7,8] developed a new approach to these problems using cocycles, and Weiss [47] proved several versions of Shannon-McMillan theorems for actions of monotileable amenable groups which we will use in this paper.…”

Section: Introductionmentioning

confidence: 99%

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On the entropy of actions of nilpotent Lie groups and their lattice subgroups

Dooley

Golodets

2011

Ergod. Th. Dynam. Sys.

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We consider a natural class $\mathcal {ULG}$ of connected, simply connected nilpotent Lie groups which contains ℝn, the group $\mathcal {UT}_n(\mathbb {R})$ of all triangular unipotent matrices over ℝ and many of its subgroups, and is closed under direct products. If $G \in \mathcal {ULG}$, then $\Gamma _1 = G\cap \mathcal {UT}_n(\mathbb {Z})$ is a lattice subgroup of G. We prove that if $G \in \mathcal {ULG}$ and Γ is a lattice subgroup of G, then a free ergodic measure-preserving action T of G on a probability space (X,ℬ,μ) has completely positive entropy (CPE) if and only if the restriction TΓ of T to Γ has CPE. We can deduce from this the following version of a well-known conjecture in this case: the action T has CPE if and only if T is uniformly mixing. Moreover, such T has a Lebesgue spectrum with infinite multiplicity. We further consider an ergodic free action T with positive entropy and suppose TΓ is ergodic for any lattice subgroup Γ of G. This holds, in particular, if the spectrum of T does not contain a discrete component. Then we show the Pinsker algebra Π(T) of T exists and coincides with the Pinsker algebras Π(TΓ) of TΓ for any lattice subgroup Γ of G. In this case, T always has Lebesgue spectrum with infinite multiplicity on the space ℒ20(X,μ)⊖ℒ20(Π(T)) , where ℒ20(Π(T)) contains all Π(T) -measurable functions from ℒ20(X,μ) . To prove these results, we use the following formula: h(T)=∣G(Γ)∣−1hK (TΓ) , where h(T) is the Ornstein–Weiss entropy of T, hK (TΓ) is a Kolmogorov–Sinai entropy of TΓ, and the number ∣G(TΓ)∣ is the Haar measure of the compact subset G(Γ) of G. In particular, h(T)=hK (TΓ1) , and hK (TΓ1)=∣G(Γ)∣−1hK (TΓ) . The last relation is an analogue of the Abramov formula for flows.

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Section: 1mentioning

confidence: 99%

“…Now, (F n ), n ∈ N is a Følner sequence of sets in G [17]. Now, (F n ), n ∈ N is a Følner sequence of sets in G [17].…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the entropy of actions of nilpotent Lie groups and their lattice subgroups

Dooley

Golodets

2011

Ergod. Th. Dynam. Sys.

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“…We also prove that if α is a non-Bernoulli cpe action of , then α G is also non-Bernoulli and cpe. Indeed, it was shown by Cornfeld et al [3] that K -mixing is equivalent to cpe for Z-actions.Kamiński [17] extended Rokhlin and Sinai's approach to actions of Z d , d < ∞, and Golodets and Sinel'shchikov [12] proved the existence of perfect partitions for actions of the group of upper triangular matrices over Z and its subgroups. We construct such a family using refinements of the classical cutting and stacking methods.…”

mentioning

confidence: 99%

“…Kamiński [17] extended Rokhlin and Sinai's approach to actions of Z d , d < ∞, and Golodets and Sinel'shchikov [12] proved the existence of perfect partitions for actions of the group of upper triangular matrices over Z and its subgroups. However, it was demonstrated that the existence of such partitions for actions of the group Z ⊕ Z ⊕ Z ⊕ · · · is a more difficult problem which remains unresolved [18].…”

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confidence: 99%

Non-Bernoulli systems with completely positive entropy

Dooley¹,

Golodets²,

Rudolph³

et al. 2008

Ergod. Th. Dynam. Sys.

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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 ∈ (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 ...

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Some ergodic and rigidity properties of discrete Heisenberg group actions

Shi

Wang

2018

Isr. J. Math.

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The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group H. We establish the decomposition of the tangent space of any C ∞ compact Riemannian manifold M for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an H group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimensional two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic H actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.

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On the entropy theory of finitely-generated nilpotent group actions

Cited by 8 publications

References 11 publications

On the entropy of actions of nilpotent Lie groups and their lattice subgroups

On the entropy of actions of nilpotent Lie groups and their lattice subgroups

Non-Bernoulli systems with completely positive entropy

Some ergodic and rigidity properties of discrete Heisenberg group actions

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