Abstract
In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk \( \overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4}))\). This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.
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Acknowledgements
We are grateful for the referee for many useful comments which improved the readability of the text.
Funding
Open Access funding provided by University of Helsinki including Helsinki University Central Hospital. The work was supported by the Finnish Academy Coe ‘Analysis and Dynamics’ and the Finnish Academy projects 1266182, 1303765, and 1309940
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Hirviniemi, O., Prause, I. & Saksman, E. Stretching and Rotation of Planar Quasiconformal Mappings on a Line. Potential Anal 59, 337–347 (2023). https://doi.org/10.1007/s11118-021-09970-9
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DOI: https://doi.org/10.1007/s11118-021-09970-9