The \v{C}erny Conjecture for aperiodic automata

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.