Measuring cones and other thick subsets in free groups

In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they acceptý. ýWe show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoidsý. ýThese automata allow us to compute explicitly generating functionsý, ý$\lambda-$measures and Cesaro measure of thick monoidsý. ýAlso we improve the asymptotic classification of regular subsets in free groupsý.