Quasiconformal and Sobolev distortion of dimension

TL;DR

This paper reviews the evolution of research on how quasiconformal, quasisymmetric, and Sobolev maps distort metric dimensions, highlighting key theorems, results, and recent advances in the field.

Contribution

It provides a comprehensive overview of the literature on dimension distortion under various mappings and introduces recent work on interpolating dimensions and their implications.

Findings

Gehring's 1973 higher integrability theorem for quasiconformal maps

Astala's 1994 solution to the planar higher integrability conjecture

Recent extensions to interpolating dimensions and classification

Abstract

We review a selection of the literature on the distortion of metric notions of dimension under quasiconformal, quasisymmetric, and Sobolev mappings. Our story begins with Gehring's landmark 1973 higher integrability theorem for quasiconformal maps, along with its implications for the distortion of Hausdorff dimension. Astala's 1994 solution to the planar higher integrability conjecture led to renewed interest in the subject in two dimensions. We continue with results from the 2000s and 2010s on the distortion of dimension by Sobolev maps, including estimates for dimension increase for generic elements in parameterized families of subsets. In the abstract metric setting, Pansu's notion of conformal dimension provides a key quasisymmetric invariant which has been useful in a wide range of applications. We briefly review relevant facts about conformal dimension, highlighting results of interest in the Euclidean setting. We conclude with recent work of the author in collaboration with Chrontsios Garitsis and with Fraser, extending the previous theory to interpolating dimensions and providing new insight into both quasiconformal classification and conformal dimension.