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 accept‎. ‎We 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 monoids‎. ‎These automata allow us to compute explicitly generating functions‎, ‎$lambda-$measures and Cesaro measure of thick monoids‎. ‎Also we improve the asymptotic classification of regular subsets in free groups‎.