Jointly ergodic measure-preserving transformations

Daniel Berend; Vitaly Bergelson

doi:10.1007/BF02760955

Jointly ergodic measure-preserving transformations

Daniel Berend, Vitaly Bergelson

Research output: Contribution to journal › Article › peer-review

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Abstract

The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that if T₁, T₂, ..., T_s are invertible commuting measure-preserving transformations of a probability space (X, ℬ, μ) then {Mathematical expression} for any f₁, f₂, ..., f_s∈L^x (X, ℬ, μ) iff T₁×T₂×...×T_s and all the transformations T_iT_j¹, i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.

Original language	English
Pages (from-to)	307-314
Number of pages	8
Journal	Israel Journal of Mathematics
Volume	49
Issue number	4
DOIs	https://doi.org/10.1007/BF02760955
State	Published - 1 Dec 1984
Externally published	Yes

ASJC Scopus subject areas

Mathematics (all)

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AB - The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that if T 1, T 2, ..., T s are invertible commuting measure-preserving transformations of a probability space (X, ℬ, μ) then {Mathematical expression} for any f 1, f 2, ..., f s∈L x (X, ℬ, μ) iff T 1×T 2×...×T s and all the transformations T iTj 1, i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.

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Jointly ergodic measure-preserving transformations

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