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}))$ 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.

In this article we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for the for almost every point on the line the set of complex stretching exponents is contained in the disk B(1/(1 − k 4 ), k 2 /(1 − k 4 )), yielding a quadratic improvement in comparison to the known optimal estimate on a general set with Hausdorff dimension 1. Our proof is based on holomorphic motions and known dimension estimates for quasicircles. In addition we establish a lower bound for the dimension of the quasiconformal image of a 1-dimensional subset of a line.

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We consider quasiconformal mappings of the unit disk that have a planar extension which have p-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this bound. Furthermore, we also obtain bounds for the compression multifractal spectra of such mappings.