  We define a new strict and computable hierarchy for the family of automaton
semigroups, which reflects the various asymptotic behaviors of the
state-activity growth. This hierarchy extends that given by Sidki for automaton
groups, and also gives new insights into the latter. Its exponential part
coincides with a notion of entropy for some associated automata.
  We prove that the Order Problem is decidable when the state-activity is
bounded. The Order Problem remains open for the next level of this hierarchy,
that is, when the state-activity is linear. Gillibert showed that it is
undecidable in the whole family.
  The former results are implemented and will be available in the GAP package
FR developed by the first author.
