Graphs induced by iterated function systems

TL;DR

This paper investigates the properties of graphs derived from iterated function systems, showing they are hyperbolic under certain conditions and their boundaries relate to the original self-similar sets.

Contribution

It introduces a new way to analyze IFSs by constructing graphs on symbolic spaces and proves their hyperbolicity under specific conditions.

Findings

Graphs are hyperbolic when the self-similar set has positive Lebesgue measure.

Hyperbolic boundaries of these graphs are homeomorphic to the self-similar sets.

Results connect geometric properties of IFSs with graph theory concepts.

Abstract

For an iterated function system (IFS) of simillitidues, we define two graphs on the representing symbolic space. We show that if the self-similar set $K$ has positive Lebesgue measure or the IFS satisfies the weak separation condition, then the graphs are hyperbolic, moreover the hyperbolic boundaries are homeomorphic to the self-similar sets.