Non-conventional ergodic averages for several commuting actions of an amenable group

Tim Austin

Research output: Contribution to journalArticle

Abstract

Let (X, μ) be a probability space, G a countable amenable group, and (Fn)n a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages (Formula Presented) associated to a commuting tuple of μ-preserving actions T1, … Td: G↷ X and f1,.., fd ∈ L(μ). We prove that these averages always converge in ‖ ⋅ ‖ 2, and that they witness a multiple recurrence phenomenon when f1 =.. = fd = 1A for a non-negligible set A ⊆ X. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.

Original languageEnglish (US)
Pages (from-to)243-274
Number of pages32
JournalJournal d'Analyse Mathematique
Volume130
Issue number1
DOIs
StatePublished - Nov 1 2016

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Ergodic Averages
Amenable Group
Probability Space
Recurrence
Countable
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ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

Cite this

Non-conventional ergodic averages for several commuting actions of an amenable group. / Austin, Tim.

In: Journal d'Analyse Mathematique, Vol. 130, No. 1, 01.11.2016, p. 243-274.

Research output: Contribution to journalArticle

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