Multi-invariant sets on tori

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Abstract

Given a compact metric group G, we are interested in those semigroups ∑of continuous endomorphisms of G, possessing the following property: The only infinite, closed, ∑-invariant subset of G is G itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization—for the case of finitedimensional tori—of those commutative semigroups with the aforementioned property.

Original languageEnglish
Pages (from-to)509-532
Number of pages24
JournalTransactions of the American Mathematical Society
Volume280
Issue number2
DOIs
StatePublished - 1 Jan 1983
Externally publishedYes

Keywords

  • Ergodic endomorphism
  • Finite-dimensional torus
  • Invariant set
  • Minimal set
  • Semigroup of endomorphisms

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