FUNCTIONS OF MARKOV PROCESSES AND ALGEBRAIC MEASURES

We introduce a class of translation-invariant measures on the set {0, …, q−1}ℤ determined by a set of q d-dimensional matrices. They are algebraic in the sense that their densities are obtained by applying a functional to products of the defining matrices. Positivity of probabilities is assured by assuming a positivity structure on the algebra of defining matrices. Restricting attention to the usual positivity notion of positive matrix elements, a detailed analysis leads to a canonical representation theorem that solves the parametrization problem. Furthermore, we show that the class of algebraic measures coincides with the class of functions of Markov processes with finite state spaces. Our main result consists in the detailed study of the asymptotics of the conditional probabilities from which we derive a formula for the mean entropy.

Translation invariant measures which are not Hausdorff measures

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.