On the Exponential Growth of Norms in Semigroups of Linear Endomorphisms and the Hausdorff Dimension of Attractors of Projective Iterated Function Systems

Given a free finitely generated semigroup S of the (normed) set of linear maps of a real or complex vector space into itself, we provide sufficient conditions for the exponential growth of the number N(k) of elements of contained in the sphere of radius k as k -> infinity and we relate the growth rate lim(k ->infinity) logN(k)/logk to the exponent of a zeta function naturally defined on S. When V = R-2 (resp., C-2 ) and S is a semigroup of volume-preserving maps, we also relate this growth rate to the Hausdorff dimension of the attractor of the induced semigroup of automorphisms of Rp(1) (resp., Cp-1).