Some topological properties and bi-Lipschitz equivalence of graph-directed attractors

TL;DR

This paper explores topological features of graph-directed iterated function systems and establishes conditions under which different attractors are bi-Lipschitz equivalent.

Contribution

It demonstrates the existence of Lipschitz embeddings between certain graph-directed attractors and identifies conditions for bi-Lipschitz equivalence.

Findings

Existence of Lipschitz embeddings between different attractors.

Conditions under which two attractors are bi-Lipschitz equivalent.

Not all attractors on the same graph are bi-Lipschitz equivalent.

Abstract

In this paper, we discuss some topological properties of the graph-directed iterated function system (GDIFS) of injective contractions. Further, we show the existence of a Lipschitz embedding between two different bi-Lipschitz graph-directed attractors. In general, two different graph-directed attractors on the same graph are not bi-Lipschitz equivalent, but under certain conditions, we show that two graph-directed attractors are bi-Lipschitz equivalent.