Rigidity properties of ℤd-actions on tori and solenoids

Manfred Einsiedler; Elon Lindenstrauss

doi:10.1090/S1079-6762-03-00117-3

Rigidity properties of ℤ^d-actions on tori and solenoids

Manfred Einsiedler
, Elon Lindenstrauss

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

27 Scopus citations

Abstract

We show that Haar measure is a unique measure on a torus or more generally a solenoid X invariant under a not virtually cyclic totally irreducible ℤ^d-action by automorphisms of X such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the nonirreducible case. These results have applications regarding measurable factors and joinings of these algebraic ℤ^d-actions.

Original language	English
Pages (from-to)	99-110
Number of pages	12
Journal	Electronic Research Announcements of the American Mathematical Society
Volume	9
Issue number	13
DOIs	https://doi.org/10.1090/S1079-6762-03-00117-3
State	Published - 14 Oct 2003
Externally published	Yes

Keywords

Entropy
Invariant measures
Invariant σ-algebras
Joinings
Measurable factors
Solenoid automorphism
Toral automorphisms

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10.1090/S1079-6762-03-00117-3

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abstract = "We show that Haar measure is a unique measure on a torus or more generally a solenoid X invariant under a not virtually cyclic totally irreducible ℤd-action by automorphisms of X such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the nonirreducible case. These results have applications regarding measurable factors and joinings of these algebraic ℤd-actions.",

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AB - We show that Haar measure is a unique measure on a torus or more generally a solenoid X invariant under a not virtually cyclic totally irreducible ℤd-action by automorphisms of X such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the nonirreducible case. These results have applications regarding measurable factors and joinings of these algebraic ℤd-actions.

KW - Entropy

KW - Invariant measures

KW - Invariant σ-algebras

KW - Joinings

KW - Measurable factors

KW - Solenoid automorphism

KW - Toral automorphisms

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ER -

Rigidity properties of ℤ^d-actions on tori and solenoids

Abstract

Keywords

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