Quasisymmetrically minimal homogeneous perfect sets

TL;DR

This paper proves that homogeneous perfect sets are minimal for 1-dimensional quasisymmetric maps, extending previous results about Cantor sets to a broader class of fractal sets.

Contribution

It establishes the minimality of homogeneous perfect sets under quasisymmetric maps, generalizing earlier findings from Cantor sets.

Findings

Homogeneous perfect sets are minimal for 1-dimensional quasisymmetric maps.

Extension of minimality results from Cantor sets to homogeneous perfect sets.

Provides a broader understanding of fractal set behavior under quasisymmetric transformations.

Abstract

In \cite{ZW}, the notion of homogenous perfect set as a generalization of Cantor type sets is introduced. Their Hausdorff, lower box-counting, upper box-counting and packing dimensions are studied in \cite{ZW} and \cite{WW}. In this paper, we show that the homogenous perfect set be minimal for 1-dimensional quasisymmetric maps, which generalize the conclusion in \cite{MS} about the uniform Cantor cantor set to the homogenous perfect set.