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Proceedings of the American Mathematical Society

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A Wolff-Denjoy theorem
for infinitely connected Riemann surfaces

Author: Finnur Lárusson
Journal: Proc. Amer. Math. Soc. 124 (1996), 2745-2750
MSC (1991): Primary 30F25, 32H50
MathSciNet review: 1342033
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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.

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Additional 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

Keywords: Riemann surface, endomorphism, iteration, Wolff-Denjoy theorem, Martin boundary, Picard existence principle, non-tangential limit, covering space, hyperbolic group, Gromov space
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
Article copyright: © Copyright 1996 American Mathematical Society

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