Amenability, Reiter's condition and Liouville property

Chu, Cho-Ho; Li, Xin

doi:10.1016/j.jfa.2018.03.014

Cited by 4 publications

(1 citation statement)

References 32 publications

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“…The current version has been rewritten by the second author. In the meantime this conjecture was independently proved by Chu -Li [CL18] in a greater generality of semigroupoids, both in the measure and in the topological categories. Although our approach and that of [CL18] are based on the same general idea from [KV83], technically they are quite different, and our argument appears to be more straightforward.…”

mentioning

confidence: 91%

Amenability of groupoids and asymptotic invariance of convolution powers

Bühler¹,

Kaimanovich²

2020

Preprint

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The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich -Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.And though absolute justice be unattainable, as much justice as we need for all practical use is attainable by all those who make it their aim.

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mentioning

confidence: 91%

Amenability of groupoids and asymptotic invariance of convolution powers

Bühler¹,

Kaimanovich²

2020

Preprint

View full text Add to dashboard Cite

show abstract

Amenability of groupoids and asymptotic invariance of convolution powers

Bühler¹,

Kaimanovich²

2021

Topology, Geometry, and Dynamics

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The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.

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A Review of Equivalent Conditions for Amenability of Countable Discrete Groups

方

2024

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Amenability, Reiter's condition and Liouville property

Cited by 4 publications

References 32 publications

Amenability of groupoids and asymptotic invariance of convolution powers

Amenability of groupoids and asymptotic invariance of convolution powers

Amenability of groupoids and asymptotic invariance of convolution powers

A Review of Equivalent Conditions for Amenability of Countable Discrete Groups

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