  In [B] Bowen defined the growth rate of an endomorphism of a finitely
generated group and related it to the entropy of a map $f:M \mapsto M$ on a
compact manifold. In this note we study the purely group theoretic aspects of
the growth rate of an endomorphism of a finitely generated group. We show that
it is finite and bounded by the maximum length of the image of a generator. An
equivalent formulation is given that ties the growth rate of an endomorphism to
an increasing chain of subgroups. We then consider the relationship between
growth rate of an endomorphism on a whole group and the growth rate restricted
to a subgroup or considered on a quotient.We use these results to compute the
growth rates on direct and semidirect products. We then calculate the growth
rate of endomorphisms on several different classes of groups including abelian
and nilpotent.
