Duke Mathematical Journal

Harmonic measure and polynomial Julia sets

I. Binder, N. Makarov, and S. Smirnov

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

Abstract

There is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for nonsimply connected domains in the plane. Because of the close relation to conformal mapping theory, the simply connected case is much better understood, and proving the above statement would give new results concerning the properties of harmonic measure in the general case.

We establish the conjecture in the category of domains bounded by polynomial Julia sets. The idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of Teichmüller's space of the polynomial and then to apply the maximum principle.

Article information

Source
Duke Math. J., Volume 117, Number 2 (2003), 343-365.

Dates
First available in Project Euclid: 26 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1085598373

Digital Object Identifier
doi:10.1215/S0012-7094-03-11725-1

Mathematical Reviews number (MathSciNet)
MR1971297

Zentralblatt MATH identifier
1036.30017

Subjects
Primary: 37F35: Conformal densities and Hausdorff dimension
Secondary: 30C85: Capacity and harmonic measure in the complex plane [See also 31A15] 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets

Citation

Binder, I.; Makarov, N.; Smirnov, S. Harmonic measure and polynomial Julia sets. Duke Math. J. 117 (2003), no. 2, 343--365. doi:10.1215/S0012-7094-03-11725-1. https://projecteuclid.org/euclid.dmj/1085598373


Export citation

References

  • K Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37--60.
  • B Branner and J H Hubbard, The iteration of cubic polynomials, I: The global topology of parameter space, Acta Math. 160 (1988), 143--206.
  • L Carleson and T W Gamelin, Complex Dynamics, Universitext Tracts Math., Springer, New York, 1993.
  • L Carleson and P W Jones, On coefficient problems for univalent functions and conformal dimension, Duke Math. J. 66 (1992), 169--206.
  • A Douady, ``Algorithms for computing angles in the Mandelbrot set'' in Chaotic Dynamics and Fractals (Atlanta, 1985), Notes Rep. Math. Sci. Engrg. 2, Academic Press, Orlando, Fla., 1986, 155--168. \CMP858 013
  • J. Graczyk and G. Swiatek, Harmonic measure and expansion on the boundary of the connectedness locus, Invent. Math. 142 (2000), 605--629.
  • P W. Jones and T. H Wolff, Hausdorff dimension of harmonic measures in the plane, Acta Math. 161 (1988), 131--144.
  • J Kiwi, ``From the shift loci to the connectedness loci of complex polynomials'' in Complex Geometry of Groups (Olmué, Chile, 1998), Contemp. Math. 240, Amer. Math. Soc., Providence, 1999, 231--245.
  • --------, Rational rays and critical portraits of complex polynomials, preprint.
  • G. M Levin, On backward stability of holomorphic dynamical systems, Fund. Math. 158 (1998), 97--107.
  • G. M Levin and M. L Sodin, ``Polynomials with a disconnected Julia set and the Green mapping'' (in Russian) in Dynamical Systems and Complex Analysis (in Russian), ``Naukova Dumka,'' Kiev, 1992, 17--24.
  • N. G Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), 369--384.
  • --. --. --. --., Fine structure of harmonic measure, St. Petersburg Math. J. 10 (1999), 217--268.
  • R Mañé, On a theorem of Fatou, Bol. Soc. Brasil Mat. (N.S.) 24 (1993), 1--11.
  • A Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), 425--430.
  • C. T McMullen and D. P Sullivan, Quasiconformal homeomorphisms and dynamics, III: The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), 351--395.
  • J Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg, Braunschweig, 1999.
  • --. --. --. --., ``Local connectivity of Julia sets: Expository lectures'' in The Mandelbrot Set, Theme and Variations, London Math. Soc. Lecture Note Ser. 274, Cambridge Univ. Press, Cambridge, 2000, 67--116.
  • --. --. --. --., ``Periodic orbits, external rays and the Mandelbrot set: An expository account'' in Geometrie complexe et systèmes dynamiques (Orsay, 1995), Asterisque 261, Soc. Math. France, Montrouge, 2000, 277--333.
  • F Przytycki, M Urbański, and A Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I, Ann. of Math. (2) 130 (1989), 1--40.
  • D Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia Math. Appl. 5, Addison-Wesley, Reading, Mass., 1978.
  • D Schleicher, On fibers and local connectivity of compact sets in $\mathbb C$, preprint.
  • S Smirnov, Symbolic dynamics and Collet-Eckmann conditions, Internat. Math. Res. Notices 2000, 333--350.