Author:
Hirviniemi Olli,Prause István,Saksman Eero
Abstract
AbstractIn 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}))$
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.
Funder
Finnish Academy Center of Excellence ’Analysis and Dynamics’
Finnish Academy
University of Helsinki including Helsinki University Central Hospital
Publisher
Springer Science and Business Media LLC
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