## Weakly mixing but not mixing quasi-Markovian processes

### Volume 142 / 2000

Studia Mathematica 142 (2000), 235-244
DOI: 10.4064/sm-142-3-235-244

#### Abstract

Let (f,α) be the process given by an endomorphism f and by a finite partition $α = {A_i}_{i=1}^{s}$ of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if $E(f,α) ⊂ { g: ⋁_{{B_i}_{i=1}^s} supp g = ⋃ _{i=1}^{s} A_{i} × B_i}$. We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the 'chequer-wise' partition for σ × S, where σ is a Bernoulli shift and S is a weakly mixing automorphism, consists of constants.