Harmonic measure and polynomial Julia sets
Harmonic measure and polynomial Julia sets
复制标题
调和测度和多项式 Julia 集
DOI：
10.1215/s0012-7094-03-11725-1
复制
发表时间：
2003
影响因子：
2.5
通讯作者：
S. Smirnov
中科院分区：
文献类型：
--
作者：
Ilia Binder;N. Makarov;S. Smirnov
有一个自然的猜想，即调和测度的维数谱的泛界对于平面上的单连通域和非单连通域是相同的。由于它与保角映射理论的密切关系，使得单连通情形得到了更好的理解，证明上述命题将给出一般情形下调和测度性质的新结果。本文在多项式Julia集有界域范畴中建立了猜想。其思想是将动态zeta函数的系数看作多项式的Teichmuller空间切片上的次调和函数，然后应用极大值原理。1.调和测度的维数谱本文讨论了复平面上调和测度的一些性质。对于定义域Ω ∈ Ω，点a ∈ Ω，设ω = ωa表示Ω在a处的调和测度.例如，测度ωa可以定义为从a开始的布朗运动的命中分布：如果e Ω，则ωa（e）是随机布朗路径在e的一点处首次命中边界的概率。当定义域尽可能一般时，人们已经做了大量的工作来描述ω的维数性质。特别地，Jones和Wolff [7]证明了，无论定义域Ω是什么，调和测度都集中在Hausdorff维数至多为1的Borel集上，即对所有平面定义域dimω ≤ 1。(1.1)我们感兴趣的是找到类似的（但更强的）普遍估计涉及的维数谱的ω。1.1.宇宙光谱。对于每个正的α，我们记f ω（α）= dim{αω（z）≤ α}，其中αω（z）是ω：αω（z）= lim inf δ→0 logωB（z，δ）log δ的点态下维数. B（z，δ）是中心为z、半径为δ的圆盘的一般符号。泛维数谱是函数Φ（α）= sup ω f ω（α），（1.2），其中上确界取在所有平面域的调和测度上。第一作者由N. S.F. Grant DMS-9970283支持。第二作者由N. S.F. Grant DMS-9800714支持。
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 Teichmuller’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.