Rigidity Theorem of Graph-directed Fractals

TL;DR

This paper establishes a rigidity theorem for dust-like graph-directed fractals, showing they are uniquely determined by Hausdorff dimension and characterizing biLipschitz equivalence within certain self-similar sets.

Contribution

It proves a rigidity theorem for graph-directed fractals with fixed ratios and integer characteristics, linking biLipschitz equivalence to Hausdorff dimension.

Findings

Dust-like graph-directed sets are rigidly determined by their Hausdorff dimensions.

Two totally disconnected self-similar sets without overlaps are biLipschitz equivalent.

BiLipschitz equivalence is characterized by Hausdorff dimension in the studied class.

Abstract

In this paper, we identify two fractals if and only if they are biLipschitz equivalent. Fix ratio $r,$ for dust-like graph-directed sets with ratio $r$ and integer characteristics, we show that they are rigid in the sense that they are uniquely determined by their Hausdorff dimensions. Using this rigidity theorem, we show that in some class of self-similar sets, two totally disconnected self-similar sets without complete overlaps are biLipschitz equivalent.