Automatic logarithm and associated measures

We introduce the notion of the automatic logarithm LogA(B) of a finite initial Mealy automaton B, with another automaton A as the base. It allows one to find for any input word w a power n such that B(w)=An(w). The purpose is to study the expanding properties of graphs describing the action of the group generated by A and B on input words of a fixed length interpreted as levels of a regular d-ary rooted tree T. Formally, the automatic logarithm is a single map LogA(B):∂T→Zd from the boundary of the tree to the d-adic integers. Under the assumption that theaction of the automaton A on the tree T is level-transitive andof bounded activity, we show that LogA(B) can be computed bya Moore machine. The distribution of values of the automatic logarithm yields a probabilistic measure μ on ∂T, which in some cases can be computed by a Mealy-type machine (we then say that μ is finite-state). We provide a criterion to determine whether μ is finite-state. A number of examples with A being the adding machine are considered.