Duke Mathematical Journal
- Duke Math. J.
- Volume 117, Number 2 (2003), 343-365.
Harmonic measure and polynomial Julia sets
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.
Duke Math. J., Volume 117, Number 2 (2003), 343-365.
First available in Project Euclid: 26 May 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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