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Journal Article

Harmonic, Gibbs and Hausdorff Measures on Repellers for Holomorphic Maps, I

Feliks Przytycki, Mariusz Urbanski and Anna Zdunik
Annals of Mathematics
Second Series, Vol. 130, No. 1 (Jul., 1989), pp. 1-40
Published by: Annals of Mathematics
DOI: 10.2307/1971475
Stable URL: http://www.jstor.org/stable/1971475
Page Count: 40
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Harmonic, Gibbs and Hausdorff Measures on Repellers for Holomorphic Maps, I
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Abstract

We prove that for a simply connected domain $\Omega \subset C$ whose boundary ∂ Ω is self-similar there is the following dichotomy, concerning the harmonic measure ω on ∂ Ω viewed from Ω: Either ∂ Ω (piecewise) real-analytic or else ω is singular with respect to the Hausdorff measure Λ Φ c (notation ω ⊥ Λ Φc ) using Makarov's function $\Phi_c(t) = t \operatorname{exp}(c\sqrt{\operatorname{log} 1/t \operatorname{log} \operatorname{log} \operatorname{log} 1/t)}$ for some $c = c(\omega) > 0$, and ω is absolutely continuous with respect to \Lambda_{\Phi_c} (notation \omega \ll \Lambda {\Phi_c)} for every c > c(\omega). So if \Omega has "fractal" boundary then the boundary compression and the "radial growth" of \vert log \vert R' \vert \vert for a Riemann mapping R: \operatorname{D} \rightarrow \Omega are as strong, respectively as fast, as permitted by Makarov's theory. We prove that c(\omega) = \sqrt{2\sigma^2/\chi for some asymptotic variance \sigma^2 for a sequence of weakly dependent random variables and a Lyapunov characteristic exponent \chi. This includes the case where \partial \Omega is a mixing repeller (in Ruelle's sense) for a holomorphic map f defined on its neighbourhood, the case \partial \Omega is a quasi-circle, invariant under the action of a quasi-Fuchsian group (for a pair of isomorphic, compact surface, Fuchsian groups) and the cases of the boundary of the "snowflake" and, more generally of Carleson's "fractal" Jordan curves. This dichotomy is partially deduced from the dichotomy concerning Gibbs measures for Holder continuous functions on an arbitrary mixing repeller X \subset C for a holomorphic map. Either \mu \bot \Lambda_k where k is the Hausdorff dimension of \mu and moreover \mu \bot \Lambda {\Phi} {(k) \atop c (\mu)} and \mu \ll \Lambda {\Phi^{(k)}_c for every c > c(\mu) for \Phi^{(k)}_c(t) = t^k \operatorname{exp} c\sqrt{\operatorname{log} 1/t \operatorname{log} \operatorname{log} \operatorname{log} 1/t}, or mu is equivalent to \Lambda_{HD(X)}. Most of the theory is carried out in the technically much harder situation of a "quasi-repeller" X--a limit set of a "tree" of pre-images of a point under iterations of a holomorphic map--and for Gibbs measures transported from the shift space to X with the use of the "tree". This includes the example X = \partial\Omega, \mu = \omega, for \Omega$ a basin of attraction to a sink for a holomorphic map. The clue is the Refined Volume Lemma which is a refinement of L.-S. Young's Volume Lemma in Pesin theory, in the sense that the Strong Law of Large Numbers is replaced by the Law of the Iterated Logarithm.

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