Invariant measures for skew products and uniformly distributed sequences

Panagiotis Georgopoulos, Constantinos Gryllakis

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


'Almost all sequences (r 1, ..., r n, ...) of positive integers have the following "universal" property: Whenever (X, μ) is a Borel probability compact metric space, and Φ 1, Φ 2, ..., Φ n, ... a sequence of commuting measure preserving continuous maps on (X, μ), such that the action (by composition) on (X, μ) of the semigroup with generators Φ 1, ..., Φ n, ... is uniquely ergodic and equicontinuous, then for every x ∈ X the sequence w 1,w 2, ..., w n, ... where, is uniformly distributed for μ. This is a contribution to Problem 116 of Schreier and Ulam in the Scottish Book.

Original languageEnglish (US)
Pages (from-to)81-103
Number of pages23
JournalMonatshefte fur Mathematik
Issue number1
StatePublished - Jul 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


  • Borel probability measure
  • Invariant measure
  • Skew product
  • Two-sided Bernoulli shift
  • Uniformly distributed sequence
  • Uniquely ergodic and equicontinuous action


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