Bilipschitz and quasiconformal rotation, stretching and multifractal spectra
Astala, Kari1 ; Iwaniec, Tadeusz1, 2 ; Prause, István1 ; Saksman, Eero1
1 Department of Mathematics and Statistics, University of Helsinki P.O. Box 68 00014 Helsinki Finland
2 Department of Mathematics, Syracuse University 13244 Syracuse NY USA
Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 113-154.

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.

DOI : 10.1007/s10240-014-0065-6
Mots-clés : Hausdorff Dimension, Quasiconformal Mapping, Multifractal Spectrum, Beltrami Equation, Logarithmic Spiral
Astala, Kari 1 ; Iwaniec, Tadeusz 1, 2 ; Prause, István 1 ; Saksman, Eero 1

1 Department of Mathematics and Statistics, University of Helsinki P.O. Box 68 00014 Helsinki Finland
2 Department of Mathematics, Syracuse University 13244 Syracuse NY USA 
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     title = {Bilipschitz and quasiconformal rotation, stretching and multifractal spectra},
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Astala, Kari; Iwaniec, Tadeusz; Prause, István; Saksman, Eero. Bilipschitz and quasiconformal rotation, stretching and multifractal spectra. Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 113-154. doi : 10.1007/s10240-014-0065-6. http://archive.numdam.org/articles/10.1007/s10240-014-0065-6/

[1.] Astala, K. Area distortion of quasiconformal mappings, Acta Math., Volume 173 (1994), pp. 37-60 | DOI | MR | Zbl

[2.] Astala, K.; Iwaniec, T.; Martin, G. J. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (2009) | Zbl

[3.] Astala, K.; Iwaniec, T.; Prause, I.; Saksman, E. Burkholder integrals, Morrey’s problem and quasiconformal mappings, J. Am. Math. Soc., Volume 25 (2012), pp. 507-531 | DOI | MR | Zbl

[4.] Balogh, Z. M.; Fässler, K.; Platis, I. D. Modulus of curve families and extremality of spiral-stretch maps, J. Anal. Math., Volume 113 (2011), pp. 265-291 | DOI | MR | Zbl

[5.] Bañuelos, R. The foundational inequalities of D.L. Burkholder and some of their raminifications, Dedicated to Don Burkholder, Ill. J. Math., Volume 54 (2010), pp. 789-868 (2012) | Zbl

[6.] Binder, I. Phase transition for the universal bounds on the integral means spectrum, Nonlinearity, Volume 22 (2009), pp. 1857-1867 | DOI | MR | Zbl

[7.] D. L. Burkholder, Sharp inequalities for martingales and stochastic integrals. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque No. 157–158 (1988), 75–94.

[8.] Evans, L. C.; Gariepy, R. F. Measure Theory and Fine Properties of Functions (1992) | Zbl

[9.] Falconer, K. Fractal Geometry. Mathematical Foundations and Applications (2003) | DOI | Zbl

[10.] Fletcher, A.; Markovic, V. Decomposing diffeomorphisms of the sphere, Bull. Lond. Math. Soc., Volume 44 (2012), pp. 599-609 | DOI | MR | Zbl

[11.] Freedman, M. H.; He, Z.-X. Factoring the logarithmic spiral, Invent. Math., Volume 92 (1988), pp. 129-138 | DOI | MR | Zbl

[12.] Gutlyanskiĭ, V.; Martio, O. Rotation estimates and spirals, Conform. Geom. Dyn., Volume 5 (2001), pp. 6-20 | DOI | MR | Zbl

[13.] Hamilton, D. H. BMO and Teichmüller space, Ann. Acad. Sci. Fenn., Ser. A 1 Math., Volume 14 (1989), pp. 213-224 | DOI | Zbl

[14.] Iwaniec, T. Nonlinear Cauchy-Riemann operators in Rn, Trans. Am. Math. Soc., Volume 354 (2002), pp. 1961-1995 | DOI | MR | Zbl

[15.] John, F. Rotation and strain, Commun. Pure Appl. Math., Volume 14 (1961), pp. 391-413 | DOI | Zbl

[16.] John, F.; Nirenberg, L. On functions of bounded mean oscillation, Commun. Pure Appl. Math., Volume 14 (1961), pp. 415-426 | DOI | MR | Zbl

[17.] Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. Éc. Norm. Super. (4), Volume 16 (1983), pp. 193-217 | Numdam | Zbl

[18.] Olsen, L. A multifractal formalism, Adv. Math., Volume 116 (1995), pp. 82-196 | DOI | MR | Zbl

[19.] C. Pommerenke, Univalent functions. Studia Mathematica, Band XXV, Göttingen, 1975.

[20.] Prause, I.; Smirnov, S. Quasisymmetric distortion spectrum, Bull. Lond. Math. Soc., Volume 43 (2011), pp. 267-277 | DOI | MR | Zbl

[21.] Reimann, H. M. On the Parametric Representation of Quasiconformal Mappings (1976), pp. 421-428

[22.] Slodkowski, Z. Holomorphic motions and polynomial hulls, Proc. Am. Math. Soc., Volume 111 (1991), pp. 347-355 | DOI | MR | Zbl

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