On the quasisymmetric minimality of homogeneous perfect sets

TL;DR

This paper investigates the quasisymmetric minimality of homogeneous perfect sets with Hausdorff dimension 1, establishing conditions under which these sets are minimal in a quasisymmetric sense.

Contribution

It introduces conditions for homogeneous perfect sets with dimension 1 to be quasisymmetrically minimal, extending understanding of their geometric properties.

Findings

Homogeneous perfect sets with dimension 1 can be quasisymmetrically minimal under certain conditions.

The paper generalizes previous results on Cantor sets to a broader class of homogeneous perfect sets.

Provides a framework for analyzing minimality in the context of quasisymmetric mappings.

Abstract

Z. Wen and J. Wu introduced the notion of homogeneous perfect sets as a generalization of Cantor type sets and determined their exact Hausdorff dimension based on the length of their fundamental intervals and the gaps between them. In this paper, we considered the minimality of the homogeneous perfect sets with Hausdorff dimension 1 and proved they are 1-dimensional quasisymmetrically minimal under some conditions.