Gromov Hyperbolic Graphs Arising From Iterations

TL;DR

This paper introduces expansive hyperbolic graphs derived from iterated function systems, exploring their properties and connections to fractal analysis, including weighted IFS and self-similar energy forms.

Contribution

It formulates a broad class of hyperbolic graphs capturing key properties of augmented trees and studies their relation to weighted IFS and fractal energy forms.

Findings

Expansive hyperbolic graphs exhibit properties like geodesics, bounded degree, and metric doubling.

The connection between weighted IFS and self-similar energy forms is established.

Hyperbolic boundaries are H"older equivalent to fractal attractors.

Abstract

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is H\"older equivalent to the attractor $K$. This setup was taken up to study the probabilistic potential theory on $K$, and the bi-Lipschitz equivalence on $K$. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and H\"older equivalence). We also study a new setup of "weighted" IFS and investigate its connection with the self-similar energy form in the analysis of fractals.