Analysis on Graphs and Fractals

Self-similar systems and their complex dimensions

E Pearse, Cornell University, USA

Abstract

Hutchinson showed that an iterated function system $\Phi$ consisting of contractive similarity mappings has a unique attractor F in d-dimensional Euclidean space, and that F is invariant under the action of the system: \Phi(F)=F. We show how the action of the function system naturally produces a tiling T of the convex hull of the attractor F. These tiles form a collection of sets whose geometry is typically much simpler than that of F, yet retains key information about both F and $\Phi$. In particular, the tiles encode all the scaling data of $\Phi$. We give the construction, along with some examples and applications. The tiling T is the foundation for the higher-dimensional extension of the theory of complex dimensions which was developed by Lapidus and van Frankenhuijsen for the case d = 1. If time permits, we will elaborate on this and give some connections to geometric measure theory/integral geometry.