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 non-simply 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. 1. Dimension spectrum of harmonic measure In this paper we discuss some properties of harmonic measure in the complex plane. For a domain Ω ⊂ Ĉ and a point a ∈ Ω, let ω = ωa denote the harmonic measure of Ω evaluated at a. The measure ωa can be defined, for instance, as the hitting distribution of a Brownian motion started at a: if e ⊂ ∂Ω, then ωa(e) is the probability that a random Brownian path first hits the boundary at a point of e. Much work has been devoted to describing dimensional properties of ω when the domain is as general as possible. In particular, Jones and Wolff [7] proved that no matter what the domain Ω is, harmonic measure is concentrated on a Borel set of Hausdorff dimension at most one; in other words, dimω ≤ 1 for all plane domains. (1.1) We are interested in finding similar (but stronger) universal estimates involving the dimension spectrum of ω. 1.1. Universal spectrum. For every positive α, we denote f ω (α) = dim{αω(z) ≤ α}, where αω(z) is the lower pointwise dimension of ω: αω(z) = lim inf δ→0 logωB(z, δ) log δ . B(z, δ) is a general notation for the disc with center z and radius δ. The universal dimension spectrum is the function Φ(α) = sup ω f ω (α), (1.2) where the supremum is taken over harmonic measures of all planar domains. The first author is supported by N.S.F. Grant DMS-9970283. The second author is supported by N.S.F. Grant DMS-9800714.