  Given a finitely generated semigroup S of the (normed) set of linear maps of
a vector space V into itself, we find sufficient conditions for the exponential
growth of the number N(k) of elements of the semigroup contained in the sphere
of radius k as k->infinity. We relate the growth rate lim log N(k)/log k to the
exponent of a zeta function naturally defined on the semigroup and, in case S
is a semigroup of volume-preserving automorpisms, to the Hausdorff and box
dimensions of the limit set of the induced semigroup of automorphisms on the
corresponding projective space.
