Topological Conformal Dimension

TL;DR

This paper introduces the topological conformal dimension, a quasisymmetrically invariant metric that bounds the topological Hausdorff dimension of images, explores its properties, and computes it for classical fractals.

Contribution

It defines and analyzes the topological conformal dimension, a new invariant, including its behavior under products, unions, and its fractional values, with computations for classical fractals.

Findings

The topological conformal dimension is quasisymmetrically invariant.

It can take fractional values.

The dimension's behavior under products and unions is characterized.

Abstract

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of quasisymmetric images of the space. We obtain results concerning the behavior of this quantity under products and unions, and compute it for some classical fractals. The range of possible values of the topological conformal dimension is also considered, and we show that this quantity can be fractional.