  We introduce the notion of `core growth rate' for strongly synchronizing
transducers. We explore some elementary properties of the core growth rate and
give examples of transducers with exponential core growth rate. We conjecture
that all strongly synchronizing transducers which generate an automaton group
of infinite order have exponential core growth rate. There is a connection to
the group of automorphisms of the one-sided shift. More specifically, the
results of this article are related to the question of whether or not there can
exist infinite order automorphisms of the one-sided shift with infinitely many
roots.
