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Harmonic measure versus Hausdorff measures on repellers for holomorphic maps

Author: Anna Zdunik
Journal: Trans. Amer. Math. Soc. 326 (1991), 633-652
MSC: Primary 58F11; Secondary 30D05, 31A15
MathSciNet review: 1031980
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Abstract: This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbański. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the Hausdorff measure corresponding to the function $ {\phi _c}(t)= t\;\exp \,\left(c\sqrt {\log \frac{1} {t}\,\log \,\log \,\log } \frac{1} {t}\right)$ for some $ c > 0$.

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  • [B] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
  • [D] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • [Go] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • [Ma] Anthony Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), no. 2, 425–430. MR 740898, 10.2307/2007044
  • [Mkl] N. G. Makarov, Determining subsets, support of a harmonic measure and perturbations of the spectrum of operators in a Hilbert space, Dokl. Akad. Nauk SSSR 274 (1984), no. 5, 1033–1037 (Russian). MR 734937
  • [Mk2] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384. MR 794117, 10.1112/plms/s3-51.2.369
  • [Ø] Bernt Øksendal, Brownian motion and sets of harmonic measure zero, Pacific J. Math. 95 (1981), no. 1, 179–192. MR 631668
  • [P1] Feliks Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), no. 1, 161–179. MR 784535, 10.1007/BF01388554
  • [P2] Feliks Przytycki, Riemann map and holomorphic dynamics, Invent. Math. 85 (1986), no. 3, 439–455. MR 848680, 10.1007/BF01390324
  • [P3] -, On holomorphic perturbations of $ z \to {z^n}$, Bull. Polish Acad. Sci. Math. 34 (1986).
  • [Ph-St] W. Phillipp and W. Stout, Almost sure invariant principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc., no. 161, 1975.
  • [PUZ] Feliks Przytycki, Mariusz Urbański, and Anna Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, Ann. of Math. (2) 130 (1989), no. 1, 1–40. MR 1005606, 10.2307/1971475
  • [Su] D. Sullivan, Seminar on conformal and hyperbolic geometry by D. P. Sullivan (Notes by M. Baker and J. Seade), preprint IHES, 1982.
  • [Z] Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, 10.1007/BF01234434

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Article copyright: © Copyright 1991 American Mathematical Society