Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

Anatole Katok, Jean Paul Thouvenot

Research output: Contribution to journalArticle

44 Scopus citations

Abstract

We define invariants for measure-preserving actions of discrete amenable groups which characterize various subexponential rates of growth for the number of "essential" orbits similarly to the way entropy of the action characterizes the exponential growth rate. We obtain above estimates for these invariants for actions by diffeomorphisms of a compact manifold (with a Borel invariant measure) and, more generally, by Lipschitz homeomorphisms of a compact metric space of finite box dimension. We show that natural cutting and stacking constructions alternating independent and periodic concatenation of names produce ℤ2 actions with zero one-dimensional entropies in all (including irrational) directions which do not allow either of the above realizations.

Original languageEnglish (US)
Pages (from-to)323-338
Number of pages16
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume33
Issue number3
DOIs
StatePublished - Jan 1 1997

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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