Exponential growth of norms in semigroups of linear automorphisms and Hausdorff dimension of self-projective IFS

TL;DR

This paper investigates the exponential growth rates of semigroups of linear automorphisms and connects these rates to the Hausdorff dimension of limit sets in projective space, providing new insights into their geometric and dynamical properties.

Contribution

It establishes sufficient conditions for exponential growth in semigroups of linear maps and relates this growth to zeta functions and Hausdorff dimensions of limit sets.

Findings

Growth rate of semigroup elements is linked to a zeta function exponent.

Exponential growth conditions are identified for semigroups of volume-preserving automorphisms.

Hausdorff and box dimensions of limit sets are characterized in relation to growth rates.

Abstract

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.