On Hyperbolic graphs induced by iterated function systems

TL;DR

This paper demonstrates that for any contractive iterated function system, a natural hyperbolic graph can be constructed on the symbolic space, establishing a Hölder equivalence between the hyperbolic boundary and the invariant set, and linking graph properties to separation conditions.

Contribution

It generalizes previous results by removing extra conditions, includes Moran sets, and connects graph degree properties to separation conditions and Lipschitz equivalence of self-similar sets.

Findings

Hyperbolic graph on symbolic space for any contractive IFS

Hölder equivalence between hyperbolic boundary and invariant set

Bounded degree property characterizes separation conditions

Abstract

For any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic space, which yields the H\"{o}lder equivalence of the hyperbolic boundary and the invariant set of the IFS. This completes the previous studies (\cite {[Ka]}, \cite{[LW1]}, \cite{[W]}) by eliminating superfluous conditions, and admits more classes of sets (e.g., the Moran sets). We also show that the bounded degree property of the graph can be used to characterize certain separation properties of the IFS (open set condition, weak separation condition); the bounded degree property is particularly important when we consider random walks on such graphs. This application and the other application to Lipschitz equivalence of self-similar sets will be discussed.