## Abstract

The problem under consideration is: when is a Markov endomorphism (onesided shift) T = T_{P} with transition matrix P, isomorphic to a Bernoulli endomorphism T̃_{ρ} 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^{-1}ε with distr δ = ρ. In this case the cofiltration (decreasing sequence of measurable partitions) ξ(T) = {T^{-n}ε}^{∞}_{n=1} generated by T is finitely isomorphic to the standard Bernoulli cofiltration ξ(T̃_{ρ}) = {T̃^{-n}_{ρ}ε}^{∞}_{n=1} and T is called finitely ρ-Bernoulli. We show that every ergodic Markov endomorphism T_{P}, which is finitely Bernoulli, can be represented as a skew product over T̃_{ρ} with d-point fibres (d ∈ ℕ). We compute the minimal d = d(T_{P}) in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which d(T_{P}) = 1, i.e. T_{P} is isomorphic to T̃_{ρ}. The cofiltration ξ(T) of any finitely Bernoulli ergodic Markov endomorphism T = T_{P} is represented as a d-point extension of the standard cofiltration ξ(T̃_{ρ}), and we show that the minimal d = d_{ξ}(T) in these extensions is equal to d(T). In particular, d(T) = 1 ⇔ d_{ξ}(T) = 1, that is, a Markov endomorphism T is isomorphic to T̃_{ρ} iff ξ(T) is isomorphic to the Bernoulli cofiltration ξ(T̃_{ρ}).

Original language | English |
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Pages (from-to) | 1527-1564 |

Number of pages | 38 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 19 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jan 1999 |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics