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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Harmonic measure versus Hausdorff measures on repellers for holomorphic maps
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by Anna Zdunik PDF
Trans. Amer. Math. Soc. 326 (1991), 633-652 Request permission

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 $c > 0$.
References
  • 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
  • Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • 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
  • Anthony Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), no. 2, 425–430. MR 740898, DOI 10.2307/2007044
  • 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
  • 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, DOI 10.1112/plms/s3-51.2.369
  • Bernt Øksendal, Brownian motion and sets of harmonic measure zero, Pacific J. Math. 95 (1981), no. 1, 179–192. MR 631668
  • 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, DOI 10.1007/BF01388554
  • Feliks Przytycki, Riemann map and holomorphic dynamics, Invent. Math. 85 (1986), no. 3, 439–455. MR 848680, DOI 10.1007/BF01390324
  • —, On holomorphic perturbations of $z \to {z^n}$, Bull. Polish Acad. Sci. Math. 34 (1986). W. Phillipp and W. Stout, Almost sure invariant principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc., no. 161, 1975.
  • 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, DOI 10.2307/1971475
  • D. Sullivan, Seminar on conformal and hyperbolic geometry by D. P. Sullivan (Notes by M. Baker and J. Seade), preprint IHES, 1982.
  • Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, DOI 10.1007/BF01234434
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 326 (1991), 633-652
  • MSC: Primary 58F11; Secondary 30D05, 31A15
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1031980-0
  • MathSciNet review: 1031980