  A deterministic finite automaton is synchronizing if there exists a word that
sends all states of the automaton to the same state. \v{C}ern\'y conjectured in
1964 that a synchronizing automaton with $n$ states has a synchronizing word of
length at most $(n-1)^2$. We introduce the notion of aperiodically
$1-$contracting automata and prove that in these automata all subsets of the
state set are reachable, so that in particular they are synchronizing.
Furthermore, we give a sufficient condition under which the \v{C}ern\'y
conjecture holds for aperiodically $1-$contracting automata. As a special case,
we prove some results for circular automata.
