A note on sensitivity of semigroup actions

Eduard Kontorovich, Michael Megrelishvili

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61 Scopus citations

Abstract

It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey (Am. Math. Mon. 99, 332-334, 1992), Silverman (Rocky Mt. J. Math. 22, 353-375, 1992) and Glasner and Weiss (Nonlinearity 6, 1067-1075, 1993). In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner (Ergodic Theory via Joinings, 2003). Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.

Original languageEnglish
Pages (from-to)133-141
Number of pages9
JournalSemigroup Forum
Volume76
Issue number1
DOIs
StatePublished - Jan 2008

Keywords

  • Almost equicontinuous
  • Almost periodic point
  • Semigroup action
  • Sensitivity
  • Transitive point

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