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

Abstract

Given a free finitely generated semigroup $S$ of the (normed) set of linear maps of a real or complex vector space $V$ into itself, we provide sufficient conditions for the exponential growth of the number $N(k)$ of elements of $S$ contained in the sphere of radius $k$ as $k\rightarrow \infty$ and we relate the growth rate $\lim _{k\rightarrow \infty }\log N(k)/\log k$ to the exponent of a zeta function naturally defined on $S$. When $V=\mathbb {R}^2$ (resp., $\mathbb {C}^2$) 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 $\mathbb {R}P^1$ (resp., $\mathbb {C}P^1$).