On Pinsker Factors for Rokhlin Entropy

Alpeev, Andrei

doi:10.1007/s10958-015-2529-8

Cited by 6 publications

(11 citation statements)

References 6 publications

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“…We have elected to remove this section from the current version of this article, as we are actually going to prove more general results in [19] which will give structural results of a probability measure-preserving action of a sofic group relative to its Pinsker factor. The results in [1] imply that Rohklin entropy decreases for compact extensions (and our results in [19] prove the same result for sofic entropy) and it appears that it is unknown whether or not naive entropy decreases under compact extensions.…”

Section: Introductionsupporting

confidence: 81%

“…We mention that completely positive Rokhlin entropy can be defined in a similar manner. After the appearance of our preprint, Alpeev in [1] proved that actions with completely positive Rokhlin entropy are weakly mixing. His approach is completely elementary.…”

Section: Corollary 12 Was Proved Formentioning

confidence: 99%

“…In a previous version of this article, we used our techniques to prove that distal measure-theoretic action have entropy zero. After the appearance of this version, Alpeev proved in [1] that measured distal actions of an arbitrary group have Rokhlin entropy zero. Then, Burton proved in [11] that distal actions have naive entropy zero if the group contains a copy of Z.…”

Section: Corollary 12 Was Proved Formentioning

confidence: 99%

“…We mentioned before that Sinaǐ's factor theorem is not known for sofic groups. Note that if Γ (X, µ) factors onto a Bernoulli shift, then by (1) we have that λ ⊕∞ Γ embeds into ρ 0 Γ (X,µ) . In this manner, Theorem 1.1 may be regarded as a weak version of Sinaǐ's Factor Theorem for sofic groups.…”

Section: Introductionmentioning

confidence: 99%

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Polish Models and Sofic Entropy

Hayes¹

2016

J. Inst. Math. Jussieu

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We deduce properties of the Koopman representation of a positive entropy probability measurepreserving action of a countable, discrete, sofic group. Our main result may be regarded as a "representationtheoretic" version of Sinaǐ's Factor Theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if Γ is a nonamenable group and Γ (X, µ) is a probability measurepreserving action which is not strongly ergodic, then no action orbit equivalent to Γ (X, µ) has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

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Section: Introductionsupporting

confidence: 81%

Section: Corollary 12 Was Proved Formentioning

confidence: 99%

Section: Corollary 12 Was Proved Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polish Models and Sofic Entropy

Hayes¹

2016

J. Inst. Math. Jussieu

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“…This is obtained through the definition of important new objects: the Pinsker algebra and the outer Pinsker algebra. See also Alpeev [1]. These are deep extensions to the nonamenable case of the theorem which states, for Z actions, that the spectrum, on the orthocomplement of L 2 of the Pinsker algebra is countable Lebesgue.…”

Section: The Continuations Of the Theorymentioning

confidence: 93%

The work of Lewis Bowen on the entropy theory of non-amenable group actions

Thouvenot¹

2019

Journal of Modern Dynamics

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We present the achievements of Lewis Bowen, or, more precisely, his breakthrough works after which a theory started to develop. The focus will therefore be made here on the isomorphism problem for Bernoulli actions of countable non-amenable groups which he solved brilliantly in two remarkable papers. Here two invariants were introduced, which led to many developments. DEFINITION 1.2. We say that two actions (X , A , µ,G, τ). and (Y , B, ν,G, σ) of the same group G are isomorphic if there exists an invertible measure preserving mapping Φ from (X , A , µ) to (Y , B, ν) such that σ g Φ = Φτ g for almost every x ∈ X , for all g ∈ G. In case the mapping Φ is not invertible, we say that (Y , B, ν,G, σ) is a factor of (X , A , µ,G, τ).

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Krieger’s finite generator theorem for actions of countable groups I

Seward

2018

Invent. math.

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We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért-Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ii) Bernoulli shifts have completely positive Rokhlin entropy; and (iii) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.h Rok G (X, µ) = inf H(α|I ) : α is a countable partition and σ-alg G (α)∨I = B(X) , where I is the σ-algebra of G-invariant sets. In this paper, we will only be interested in ergodic actions, in which case the Rokhlin entropy simplifies to h Rok G (X, µ) = inf H(α) : α is a countable generating partition .

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On Pinsker Factors for Rokhlin Entropy

Abstract: In this paper we will prove that any dynamical system posess the unique maximal factor of zero Rokhlin entropy, so-called Pinsker factor. It is proven also, that if the system is ergodic and this factor has no atoms then system is relatively weakly mixing extension of its Pinsker factor.

Cited by 6 publications

References 6 publications

Polish Models and Sofic Entropy

Polish Models and Sofic Entropy

The work of Lewis Bowen on the entropy theory of non-amenable group actions

Krieger’s finite generator theorem for actions of countable groups I

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