Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 103-123

STRETCHING AND ROTATION SETS OF QUASICONFORMAL MAPPINGS

Tyler Bongers

Michigan State University, Department of Mathematics
619 Red Cedar Road, East Lansing, MI 48824, U.S.A.; charlesb 'at' math.msu.edu

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 critical 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.

2010 Mathematics Subject Classification: Primary 30C62, 28A78.

Key words: Quasiconformal mapping, Hausdorff dimension, Beltrami equation.

Reference to this article: T. Bongers: Stretching and rotation sets of quasiconformal mappings. Ann. Acad. Sci. Fenn. Math. 44 (2019), 103-123.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4423