A construction of non-isomorphic one-sided Markov shifts

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Let Tρ be a one-sided Bernoulli shift, which acts on the product space (Xρρ) = Π n=1=(I,ρ) is a finite or countable state space with a probability distribution ρ = {ρ(i)}iεI, ρ(i) > 0, Σiε ρ(i) = 1. Let also Yd = {1,2, • • • d}, d e N. Suppose p is not homogeneous, i.e. ρ(i) σ(i') for some pair t, i' ε I, and let d > 1. For given such d and p, we construct a uncountable family Tλ, λ ε A = A(ρ, d), such that (i) Tλ is isomorphic to a d-extension of Tρ, i.e. to a skew product on Xρ × Y d over Tρ. (ii) Tλ is an one-sided Markov shift, corresponding to an aperiodic irreducible positively recurrent Markov chain on the infinite countable state space I × N × Y d. (iii) All the shifts Tλ λ ε A, are pairwise non-isomorphic. (iv) Each of the shifts Tλ λ ε A, is not Isomorphic to one-sided Markov shifts on finite state spaces (even in the case, when I is finite).

Original languageEnglish
Pages (from-to)273-297
Number of pages25
JournalIndian Journal of Mathematics
Issue number3
StatePublished - 1 Dec 2012


  • One sided markov shifts
  • Skew products over bernoulli shifts isomorphism problem

ASJC Scopus subject areas

  • Mathematics (all)


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