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

23 Scopus citations

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

General Mathematics

Access to Document

10.1007/BF02760955

Cite this

@article{a26f870624b84965a62bbbc3f53e753d,

title = "Jointly ergodic measure-preserving transformations",

abstract = "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{\texttimes}T 2{\texttimes}...{\texttimes}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.",

author = "Daniel Berend and Vitaly Bergelson",

year = "1984",

month = dec,

day = "1",

doi = "10.1007/BF02760955",

language = "English",

volume = "49",

pages = "307--314",

journal = "Israel Journal of Mathematics",

issn = "0021-2172",

publisher = "Springer New York",

number = "4",

}

TY - JOUR

T1 - Jointly ergodic measure-preserving transformations

AU - Berend, Daniel

AU - Bergelson, Vitaly

PY - 1984/12/1

Y1 - 1984/12/1

N2 - 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.

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.

UR - https://www.scopus.com/pages/publications/0013076511

U2 - 10.1007/BF02760955

DO - 10.1007/BF02760955

M3 - Article

AN - SCOPUS:0013076511

SN - 0021-2172

VL - 49

SP - 307

EP - 314

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 4

ER -

Jointly ergodic measure-preserving transformations

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this