Bilipschitz and quasiconformal rotation, stretching and multifractal spectra

Abstract

We establish sharp bounds for simultaneous local rotation and Hölder-distortion of planar quasiconformal maps. In addition, we give sharp estimates for the corresponding joint quasiconformal multifractal spectrum, based on new estimates for Burkholder functionals with complex parameters. As a consequence, we obtain optimal rotation estimates also for bi-Lipschitz maps.2010 Mathematics Subject Classification. Primary 30C62; Secondary 37C45.

On multifractal spectrum of quasiconformal mappings

“…Theorem 1.1, together with the examples constructed in [2] verifying optimality, completely answers the question concerning the Hausdorff dimension of the sets E f , and hence gives the optimal dimension for sets where K-quasiconformal mapping can stretch and rotate according to given parameters. This raises a natural question whether the sharpness of the dimension in Theorem 1.1 can be extended to the sharpness on the level of Hausdorff measures.…”

“…We study the multifractal spectra of quasiconformal mappings, which means that we are interested in the maximum size of the sets in which quasiconformal mapping stretches and rotates according to given parameters. We construct examples of quasiconformal mappings which improve a previous result from [2] in the sense of Hausdorff measure. …”

“…When studying the argument of a principal quasiconformal mapping it is natural to pick the principal branch of the logarithm, as in [2]. Namely, we choose the branch so that the notion arg(f (z 0 + e iβ r 0 ) − f (z 0 )) can be understood as a rotation of f (z) = f (z 0 + e iβ r) around the point f (z 0 ), when z = z 0 + e iβ r moves from r = +∞ to r = r 0 along the line which passes trough points z 0 and z 0 + e iβ r 0 .…”

“…Given a mapping f and parameters α, γ we denote by E f = E f,α,γ the set of points that satisfy (1.1). For the Hausdorff dimension of these sets the following sharp result is given in [2]. Theorem 1.1.…”

“…As a function of the variable α(1 + iγ) the function (1.4) is determined as the unique 'cone'-like function on the disc B K that takes value 2 at the point 1, vanishes on the boundary of B K , and is linear on every line segment joining 1 to the boundary of B K , see [2] Remark 5.2. Theorem 1.1, together with the examples constructed in [2] verifying optimality, completely answers the question concerning the Hausdorff dimension of the sets E f , and hence gives the optimal dimension for sets where K-quasiconformal mapping can stretch and rotate according to given parameters.…”