The doubling metric and doubling measures

We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset $\mathbb{U}$ of a metric space $X$, the predecessor $\mathbb{U}_\ast$ of $\mathbb{U}$ is defined by doubling the radii of all open balls contained inside $\mathbb{U}$, and taking their union. The predecessor of $\mathbb{U}$ is an open set containing $\mathbb{U}$. The directed doubling distance between $\mathbb{U}$ and another subset $\mathbb{V}$ is the number of times that the predecessor operation needs to be applied to $\mathbb{U}$ to obtain a set that contains $\mathbb{V}$. Finally, the doubling distance between open sets $\mathbb{U}$ and $\mathbb{V}$ is the maximum of the directed distance between $\mathbb{U}$ and $\mathbb{V}$ and the directed distance between $\mathbb{V}$ and $\mathbb{U}$.

Keywords

metric, doubling measure, quasisymmetric map

2010 Mathematics Subject Classification

Primary 54E35. Secondary 28A12, 51F99.

Full Text (PDF format)

V.S. has been partly supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research.

Received 3 September 2019

Received revised 13 April 2020

Accepted 28 April 2020

Published 3 November 2020