论文标题
平面准文明映射在线上的拉伸和旋转
Stretching and Rotation of Planar Quasiconformal Mappings on a Line
论文作者
论文摘要
在本文中，我们研究了平面准文献映射的拉伸和旋转。我们表明，在线上几乎每个点，复杂的拉伸指数集（共同描述拉伸和旋转）都包含在磁盘$ \ overline {b}（1/（1-k^4），k^2/（1-k^4））中。这对Hausdorff Dimension $ 1 $的一般最佳估计值进行了二次改进。我们的证明是基于尸体形态运动和准圆的尺寸的估计。我们还为$ 1 $维尺寸子集的图像的尺寸提供了一个下限。
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.
