Quasi-symmetric invariant properties of Cantor metric spaces

TL;DR

This paper explores the properties of Cantor metric spaces, distinguishing between standard and exotic types based on quasi-symmetric invariants, and demonstrates the vast diversity of conformal gauges among exotic spaces.

Contribution

It classifies Cantor metric spaces into standard and exotic types and shows the continuum of conformal gauges for each exotic type, also constructing spaces with prescribed dimensions.

Findings

Exotic Cantor metric spaces have continuum many conformal gauges.

Standard spaces satisfy all three quasi-symmetric invariants.

Existence of Cantor spaces with prescribed Hausdorff and Assouad dimensions.

Abstract

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic types the class of all the conformal gauges of Cantor metric spaces has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.