Stretching and Rotation Sets of Quasiconformal Mappings

TL;DR

This paper investigates the size and measure-theoretic properties of sets where quasiconformal maps exhibit specific stretching and rotation behaviors, advancing understanding of their singularities and geometric distortion properties.

Contribution

It improves previous results by providing examples of stretching and rotation sets with non-sigma-finite measure, positive Riesz capacity, and positivity for gauged Hausdorff measures at critical dimensions.

Findings

Constructed examples with non-sigma-finite measure at Hausdorff dimension.

Demonstrated positive Riesz capacity at critical homogeneity.

Established positivity for broad classes of gauged Hausdorff measures.

Abstract

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension.