  A word w is called a synchronizing (recurrent, reset) word of a deterministic
finite automaton (DFA) if w brings all states of the automaton to some state; a
DFA that has a synchronizing word is said to be synchronizing. Cerny
conjectured in 1964 that every n-state synchronizing DFA possesses a
synchronizing word of length at most (n -1)2. We consider automaton with
aperiodic transition monoid (such automaton is called aperiodic). We show that
every synchronizing n-state aperiodic automaton has a synchronizing word of
length at most n(n-2)+1. Thus, for aperiodic automaton as well as for
automatons accepting only star-free languages, the Cerny conjecture holds true.
