Abstract
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 ℤ, 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.
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Courbage, M., Hamdan, D. A class of nonmixing dynamical systems with monotonic semigroup property. Lett Math Phys 22, 101–106 (1991). https://doi.org/10.1007/BF00405173
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DOI: https://doi.org/10.1007/BF00405173