A class of nonmixing dynamical systems with monotonic semigroup property

In order to illustrate the class of conservative dynamical systems for which a Boltzmann entropy can be obtained under finite coarse-graining 2], we consider dynamical systems defined by the shift transformation on K , where K is any finite set of integers. We give a class of non-Markovian invariant measures that verify the Chapman-Kolmogorov equation (equivalent to a Boltzmann entropy) for any positive stochastic matrix and that are ergodic but not weakly mixing.