Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Radial images by holomorphic mappings


Authors: José L. Fernández and Domingo Pestana
Journal: Proc. Amer. Math. Soc. 124 (1996), 429-435
MSC (1991): Primary 30E25, 30F45
DOI: https://doi.org/10.1090/S0002-9939-96-02971-1
MathSciNet review: 1283549
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal{R}$ be a nonexceptional Riemann surface, other than the punctured disk. We prove that if $f$ is a holomorphic mapping from the unit disk $\Delta $ of the complex plane into $\mathcal{R}$, then the set of radial images that remain bounded in the Poincaré metric of $\mathcal{R}$ has Hausdorff dimension at least $\delta (\mathcal{R})$, the exponent of convergence of $\mathcal{R}$. The result is best possible. This is a hyperbolic analog of the result of N. G. Makarov that Bloch functions are bounded on a set of radii of dimension one.


References [Enhancements On Off] (What's this?)

  • A1 L. V. Ahlfors, Conformal invariants, McGraw-Hill, New York, 1973, MR 50:10211.
  • A2 ------, Quasiconformal mappings, Van Nostrand, Princeton, NJ, 1966, MR 34:336.
  • AP J. M. Anderson and L. D. Pitt, The boundary behaviour of Bloch functions and univalent functions, Michigan Math. J. 35 (1988), 313--320, MR 89m:30067.
  • Be A. F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983, MR 85d:22026.
  • BJ C. Bishop and P. Jones, Hausdorff dimension and Kleinian groups, preprint, 1994.
  • Bo J. Bourgain, On the radial variation of bounded analytic functions on the disc, Duke Math. J. 69 (1993), 671--682, MR 94d:30061.
  • FM J. L. Fernández and M. V. Melián, Bounded geodesics of Riemann surfaces and hyperbolic manifolds, Trans. Amer. Math. Soc. 347 (1995), 3533--3549, CMP 95:02.
  • FP J. L. Fernández and D Pestana, Distortion of boundary sets under inner functions and applications, Indiana Univ. Math. J. 41 (1992), 439--447, MR 93k:30014.
  • FR J. L. Fernández and J. M. Rodríguez, The exponent of convergence of Riemann surfaces. Bass Riemann surfaces, Ann. Acad. Sci. Fenn. 15 (1990), 165--183, MR 91b:58263.
  • H D. Hamilton, Conformal distortion of boundary sets, Trans. Amer. Math. Soc. 308 (1988), 69--81, MR 90h:30020.
  • LV O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, New York, 1973, MR 49:9202.
  • M1 N. G. Makarov, On the radial behaviour of Bloch functions, Soviet Math. Dokl. 40 (1990), 505--508.
  • M2 ------, Smooth measures and the law of the iterated logarithm, Math. USSR-Izv. 34 (1990), 455--463, MR 90h:30007.
  • Ni P. Nicholls, The ergodic theory of discrete groups, London Math. Soc. Lecture Note Series, vol. 143, Cambridge Univ. Press, Cambridge, 1989, MR 91i:58104.
  • R S. Rohde, The boundary behaviour of Bloch functions, J. London Math. Soc. (2) 48 (1993), 488--499, MR 94k:30083.
  • St B. Stratmann, The Hausdorff dimension of bounded geodesics on geometrically finite manifolds, preprint, 1993.
  • T M. Tsuji, Potential theory in modern function theory, Chelsea, New York, 1959, MR 22:5712.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30E25, 30F45

Retrieve articles in all journals with MSC (1991): 30E25, 30F45


Additional Information

José L. Fernández
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: pando@ccuam3.sdi.uam.es

Domingo Pestana
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Address at time of publication: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganes, Spain
Email: dompes@arwen.uc3m.es

DOI: https://doi.org/10.1090/S0002-9939-96-02971-1
Received by editor(s): May 6, 1994
Received by editor(s) in revised form: June 16, 1994
Additional Notes: Research supported by a grant of CICYT, Ministerio de Educación y Ciencia, Spain.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society