On Mealy–Moore coding and images of Markov measures

Abstract

We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.

Automatic logarithm and associated measures

“…We then analyze the distribution of the lengths of the łchordsž (again we appeal to Figure 3 which gives an impression of what we mean by the chord). This leads us to the considerations started in the [5,11] about the nature of the image of the Bernoulli (or, more generally, Markov) measure under the automaton map, in the case the map is given by the łlogarithmž automaton L. The distribution of the chords is given by the image µ = L * (ν) of the uniform Bernoulli measure ν on X N , which in some important cases (for instance, given in Example 8.5 and Theorem 8.6) is a Markov measure, but in some other interesting cases (like Example 8.7) is a more complicated type of measure.…”