On finitely Bernoulli one-sided Markov shifts and their cofiltrations

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The problem under consideration is: when is a Markov endomorphism (onesided shift) T = TP with transition matrix P, isomorphic to a Bernoulli endomorphism T̃ρ with an appropriate stationary vector ρ = {ρi}i∈I? An obvious necessary condition is that there exists an independent complement δ of the measurable partition T-1ε with distr δ = ρ. In this case the cofiltration (decreasing sequence of measurable partitions) ξ(T) = {T-nε}n=1 generated by T is finitely isomorphic to the standard Bernoulli cofiltration ξ(T̃ρ) = {T̃-nρε}n=1 and T is called finitely ρ-Bernoulli. We show that every ergodic Markov endomorphism TP, which is finitely Bernoulli, can be represented as a skew product over T̃ρ with d-point fibres (d ∈ ℕ). We compute the minimal d = d(TP) in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which d(TP) = 1, i.e. TP is isomorphic to T̃ρ. The cofiltration ξ(T) of any finitely Bernoulli ergodic Markov endomorphism T = TP is represented as a d-point extension of the standard cofiltration ξ(T̃ρ), and we show that the minimal d = dξ(T) in these extensions is equal to d(T). In particular, d(T) = 1 ⇔ dξ(T) = 1, that is, a Markov endomorphism T is isomorphic to T̃ρ iff ξ(T) is isomorphic to the Bernoulli cofiltration ξ(T̃ρ).

Original languageEnglish
Pages (from-to)1527-1564
Number of pages38
JournalErgodic Theory and Dynamical Systems
Issue number6
StatePublished - 1 Jan 1999

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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