Quantitative measure equivalence between amenable groups

Delabie, Thiebout; Koivisto, Juhani; Maître, François Le; Tessera, Romain

doi:10.48550/arxiv.2002.00719

Cited by 5 publications

(13 citation statements)

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“…If one is permitted to use two different acting groups then one can show much more easily that certain actions which are known not to be integrably orbit equivalent are in fact Shannon orbit equivalent. This phenomenon was observed in [5] in the context of measure equivalence for groups. Consider for example the odometer Z-action on {0, 1, 2, 3} N and the action of Z 2 on {0, 1} N × {0, 1} N = ({0, 1} × {0, 1}) N implemented on the canonical generators by T × id and id × T where T denotes the odometer transformation of {0, 1} N .…”

Section: Odometersmentioning

confidence: 62%

Entropy, virtual Abelianness, and Shannon orbit equivalence

Kerr¹,

Li²

2022

Preprint

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We prove that if two free p.m.p. Z-actions are Shannon orbit equivalent then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein-Weiss and the entropy invariance results of Austin and Kerr-Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every Z-odometer is Shannon orbit equivalent to the universal Z-odometer.

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Section: Odometersmentioning

confidence: 62%

Entropy, virtual Abelianness, and Shannon orbit equivalence

Kerr¹,

Li²

2022

Preprint

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“…This fits into the emerging field of quantitative orbit equivalence for group actions. One of its tacit aims is to capture meaningful geometric invariants, such as Følner functions [DKLMT20], growth rates [Aus16b], etc., or ergodic theoretic invariants, such as dynamical entropy [Aus16a].…”

Section: Introductionmentioning

confidence: 99%

Belinskaya's theorem is optimal

Carderi¹,

Matthieu²,

Maître³

et al. 2022

Preprint

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Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an L 1 cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all p < 1 the integrability condition on the cocycle cannot be relaxed to being in L p . This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.

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“…actions of finitely generated groups via the graphings they define. The spirit of it fits into the framework of quantitative orbit equivalence and more generally quantitative measure equivalence [DKLMT20]. These nascent areas aim to understand how metric structures on orbits of p.m.p.…”

Section: Introductionmentioning

confidence: 99%

Isometric orbit equivalence for probability-measure preserving actions

Matthieu¹

2022

Preprint

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We study probability-measure preserving (p.m.p.) actions of finitely generated groups via the graphings they define. We introduce and study the notion of isometric orbit equivalence for p.m.p. actions: two p.m.p. actions are isometric orbit equivalent if the graphings defined by some fixed generating systems of the groups are measurably isometric.We highlight two kind of phenomena. First, we prove that the notion of isometric orbit equivalence is rigid for groups whose Cayley graph, with respect to a fixed generating system, has a countable group of automorphism. On the other hand, we introduce a general construction of isometric orbit equivalent p.m.p. actions, which leads to interesting nontrivial examples of isometric orbit equivalent p.m.p. actions for the free group. In particular, our examples show that mixing is not invariant under isometric orbit equivalence.

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Quantitative measure equivalence between amenable groups

Cited by 5 publications

References 0 publications

Entropy, virtual Abelianness, and Shannon orbit equivalence

Entropy, virtual Abelianness, and Shannon orbit equivalence

Belinskaya's theorem is optimal

Isometric orbit equivalence for probability-measure preserving actions

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