Log in :: RSS/Atom Feeds
Illinois Journal of Mathematics

Dimension of elliptic harmonic measure of snowspheres

Daniel Meyer
Source: Illinois J. Math. Volume 53, Number 2 (2009), 691-721.

Abstract

A metric space $\mathcal{S}$ is called a quasisphere if there is a quasisymmetric homeomorphism $f\dvtx S^{2}\to\mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by f. It is shown that for certain self similar quasispheres $\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\mathcal{S}$. This result is obtained by representing the self similarity of a snowsphere by a postcritically finite rational map, and showing a corresponding result for such maps. As a corollary a metric characterization of Lattès maps is obtained. Furthermore, a method to compute the dimension of elliptic harmonic measure numerically is presented, along with the (numerically computed) values for certain examples.

First Page: Show Hide
Primary Subjects: 30C65
Secondary Subjects: 37A05, 37F10
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ijm/1266934799
Zentralblatt MATH identifier: 05676343
Mathematical Reviews number (MathSciNet): MR2594650

References

L. V. Ahlfors and A. Beurling, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125--142.
Mathematical Reviews (MathSciNet): MR0086869
Zentralblatt MATH: 0072.29602
Digital Object Identifier: doi:10.1007/BF02392360
M. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), 127--183.
Mathematical Reviews (MathSciNet): MR1930885
Zentralblatt MATH: 1037.53023
Digital Object Identifier: doi:10.1007/s00222-002-0233-z
M. Bonk and D. Meyer, Expanding Thurston maps, in preparation.
J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), 477--483.
Mathematical Reviews (MathSciNet): MR0874032
Zentralblatt MATH: 0616.31004
Digital Object Identifier: doi:10.1007/BF01389238
J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Constructing rational maps from subdivision rules, Conform. Geom. Dyn. 7 (2003), 76--102.
Mathematical Reviews (MathSciNet): MR1992038
Digital Object Identifier: doi:10.1090/S1088-4173-03-00082-1
J. W. Cannon, W. J. Floyd and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153--196 (electronic).
Mathematical Reviews (MathSciNet): MR1875951
Digital Object Identifier: doi:10.1090/S1088-4173-01-00055-8
L. Carleson and T. W. Gamelin, Complex dynamics, Springer, New York, 1993.
Mathematical Reviews (MathSciNet): MR1230383
Zentralblatt MATH: 0782.30022
A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), 263--297.
Mathematical Reviews (MathSciNet): MR1251582
Zentralblatt MATH: 0806.30027
Digital Object Identifier: doi:10.1007/BF02392534
G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann. 315 (1999), 641--710.
Mathematical Reviews (MathSciNet): MR1731465
Zentralblatt MATH: 0944.53004
Digital Object Identifier: doi:10.1007/s002080050332
K. Falconer, Fractal geometry. Mathematical foundations and applications, Wiley, Chichester, 1990.
Mathematical Reviews (MathSciNet): MR1102677
J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math. 175 (2009), 335--415.
Mathematical Reviews (MathSciNet): MR2470110
Zentralblatt MATH: 1163.37008
Digital Object Identifier: doi:10.1007/s00222-008-0152-8
J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, New York, 2001.
Mathematical Reviews (MathSciNet): MR1800917
Zentralblatt MATH: 0985.46008
P. Haïssinsky and K. M. Pilgrim, Coarse expanding conformal dynamics, Astérisque, vol. 325, Société Mathématique de France, Paris, 2009.
S. Lattès, Sur l'itération des substitutions rationnelles et les fonctions de Poincaré, C. R. Acad. Sci. Paris 166 (1918), 26--28 (French).
N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), 369--384.
Mathematical Reviews (MathSciNet): MR0794117
Zentralblatt MATH: 0573.30029
Digital Object Identifier: doi:10.1112/plms/s3-51.2.369
A. Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), 425--430.
Mathematical Reviews (MathSciNet): MR0740898
Digital Object Identifier: doi:10.2307/2007044
JSTOR: links.jstor.org
R. Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86--117.
Mathematical Reviews (MathSciNet): MR0961095
C. T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994.
Mathematical Reviews (MathSciNet): MR1312365
Zentralblatt MATH: 0822.30002
D. Meyer, Snowballs are quasiballs, Trans. Amer. Math. Soc. 362 (2010), 1247--1300.
Mathematical Reviews (MathSciNet): MR2563729
Zentralblatt MATH: 1190.30018
Digital Object Identifier: doi:10.1090/S0002-9947-09-04635-2
--------, Quasisymmetric embedding of self similar surfaces and origami with rational maps, Ann. Acad. Sci. Fenn. Math 27 (2002), 461--484.
Mathematical Reviews (MathSciNet): MR1922201
J. Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006.
Mathematical Reviews (MathSciNet): MR2193309
--------, On Lattès maps, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 9--43.
Mathematical Reviews (MathSciNet): MR2348953
Digital Object Identifier: doi:10.4171/011-1/1
K. Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, 1983.
Mathematical Reviews (MathSciNet): MR0833286
F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), 309--317.
Mathematical Reviews (MathSciNet): MR1186141
Zentralblatt MATH: 0787.58037
Digital Object Identifier: doi:10.2307/2159858
JSTOR: links.jstor.org
F. Przytycki and M. Urbański, Conformal fractals-Ergodic Theory Methods, London Math. Soc. Lect. Note Series, vol. 371. Cambridge University Press, Cambridge, 2010.
Mathematical Reviews (MathSciNet): MR2656475
Zentralblatt MATH: 05715320
P. Tukia, Hausdorff dimension and quasisymmetric mappings, Math. Scand. 65 (1989), 152--160.
Mathematical Reviews (MathSciNet): MR1051832
Zentralblatt MATH: 0677.30016
M. Urbański, Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc. 40 (2003), 281--312.
Mathematical Reviews (MathSciNet): MR1978566
Digital Object Identifier: doi:10.1090/S0273-0979-03-00985-6
Zentralblatt MATH: 1031.37041
J. Väisälä, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathematics, vol. 229, Springer, Berlin, 1971.
Mathematical Reviews (MathSciNet): MR0454009
P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer, New York, 1982.
Mathematical Reviews (MathSciNet): MR0648108
T. H. Wolff, Counterexamples with harmonic gradients in $\bf R\sp 3$, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 321--384.
Mathematical Reviews (MathSciNet): MR1315554
Zentralblatt MATH: 0836.31004
A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), 627--649.
Mathematical Reviews (MathSciNet): MR1032883
Zentralblatt MATH: 0820.58038
Digital Object Identifier: doi:10.1007/BF01234434
M. Zinsmeister, Thermodynamic formalism and holomorphic dynamical systems, SMF/AMS Texts and Monographs, vol. 2, Amer. Math. Soc., Providence, RI, 2000.
Mathematical Reviews (MathSciNet): MR1724307
Zentralblatt MATH: 0959.37003

2013 © University of Illinois at Urbana-Champaign, Department of Mathematics

Illinois Journal of Mathematics

Illinois Journal of Mathematics

Turn MathJax Off
What is MathJax?