On the period-two-property of the majority operator in infinite graphs

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A self-mapping M: X → X of a nonempty set X has the Period-Two-Property (p2p) if M2x = x holds for every Af-periodic point x ∈ X. Let X be the set of all (0, 1) -labelings x: V →(0, 1) of the set of vertices F of a locally finite connected graph G. For x ∈ X let Mx ∈ X label v e V by the majority bit that x applies to its neighbors, retaining v’s x-label in case of a tie. We show that M has the p2p if there is a finite bound on the degrees in G and 1/n log b→0, where bnis the number of v ∈ V at a distance at most n from a fixed vertex v ∈ V.

Original languageEnglish
Pages (from-to)1649-1667
Number of pages19
JournalTransactions of the American Mathematical Society
Issue number5
StatePublished - May 1995


  • Cellular automata
  • Infinite graphs
  • Majority rule
  • Period-two

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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