There is a well-known
construction which associates with each σ-finite measure space (X,𝒮,μ) a certain
stochastic process {N(F) : F ∈𝒮,μ(F) < ∞} called the Poisson process over
(X,𝒮,μ). Any μ-preserving bimeasurable map τ on X “lifts” to a probability
preserving map T, characterized by N(F) ∘ T = N(τ−1F). We show the
following: If τ is the shift arising from a Markov chain preserving a σ-finite
measure with stochastic matrix (pi,j)i,j∈N. Then T is a Bernoulli shift iff
pi,jn→ 0∀i,j ∈ N as n →∞. If, in addition, τ has a recurrent state or if it is
transient and (𝒮,μ) is not completely atomic, then T has infinite entropy. The
analogous results are valid for ν-step Markov chains preserving a σ-finite measure
(ν > 1).
