Bilipschitz embedding of homogeneous fractals

TL;DR

This paper introduces homogeneous fractals, a class including Ahlfors-David regular sets, and investigates their dimensions, bilipschitz embeddings, and quasi-Lipschitz equivalences using Moran sets.

Contribution

It defines homogeneous fractals based on measure properties and explores their embedding and equivalence characteristics, extending understanding of irregular fractal sets.

Findings

Homogeneous fractals include all Ahlfors-David regular sets.

The paper characterizes bilipschitz embedding conditions for these fractals.

It establishes criteria for quasi-Lipschitz equivalence among homogeneous fractals.

Abstract

In this paper, we introduce a class of fractals named homogeneous sets based on some measure versions of homogeneity, uniform perfectness and doubling. This fractal class includes all Ahlfors-David regular sets, but most of them are irregular in the sense that they may have different Hausdorff dimensions and packing dimensions. Using Moran sets as main tool, we study the dimensions, bilipschitz embedding and quasi-Lipschitz equivalence of homogeneous fractals.