The Journal of Geometric Analysis

, Volume 25, Issue 3, pp 1798–1827

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


DOI: 10.1007/s12220-014-9494-1

Cite this article as:
De Leo, R. J Geom Anal (2015) 25: 1798. doi:10.1007/s12220-014-9494-1


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., \(\mathbb {C}P^1\)).


SemigroupsMatricesZeta functionsHausdorff dimensionSelf-projective setsIterated function systemsJoint spectral radius

Mathematics Subject Classification


Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Howard UniversityWashingtonUSA
  2. 2.INFNCagliariItaly