A Wolff-Denjoy theorem for infinitely connected Riemann surfaces
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- by Finnur Lárusson
- Proc. Amer. Math. Soc. 124 (1996), 2745-2750
- DOI: https://doi.org/10.1090/S0002-9939-96-03451-X
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Abstract:
We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let $X$ be a non-parabolic Riemann surface with Martin boundary $\Delta$. Suppose each Martin function $k_{y}$, $y\in \Delta$, extends continuously to $\Delta \setminus \{y\}$ and vanishes there. We show that if $f$ is an endomorphism of $X$ and the iterates of $f$ converge to the point at infinity, then the iterates converge locally uniformly to a point in $\Delta$. As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on $f$ but none on $X$ itself.References
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Bibliographic Information
- Finnur Lárusson
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Address at time of publication: Department of Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
- MR Author ID: 347171
- Email: Larusson@uwo.ca
- Received by editor(s): March 3, 1995
- Additional Notes: This work was supported in part by the Icelandic Council of Science and by the U.S. National Science Foundation under grant no. DMS-9400872.
- Communicated by: Theodore W. Gamelin
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2745-2750
- MSC (1991): Primary 30F25, 32H50
- DOI: https://doi.org/10.1090/S0002-9939-96-03451-X
- MathSciNet review: 1342033