  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.
