A ratio ergodic theorem for multiparameter non-singular actions

Michael Hochman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

We prove a ratio ergodic theorem for non-singular free ℤd and ℝd actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in ℝd. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group actions, the Besicovitch covering property not only implies the maximal inequality, but is equivalent to it, implying that further generalization may require new methods.

Original languageAmerican English
Pages (from-to)365-383
Number of pages19
JournalJournal of the European Mathematical Society
Volume12
Issue number2
DOIs
StatePublished - 2010
Externally publishedYes

Keywords

  • Commuting transformations
  • Ergodic theorem
  • Group actions
  • Maximal inequality
  • Measure preserving transformations
  • Non-singular actions

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