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Conformal Geometry and Dynamics

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On the failure of a generalized Denjoy-Wolff theorem

Author: Pietro Poggi-Corradini
Journal: Conform. Geom. Dyn. 6 (2002), 13-32
MSC (2000): Primary 30D05, 31A05
Published electronically: January 24, 2002
MathSciNet review: 1882086
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Abstract: The classical Denjoy-Wolff Theorem for the unit disk was generalized, in 1988, by Maurice Heins, to domains bounded by finitely many analytic Jordan curves. Heins asked whether such an extension is valid more generally. We show that it can actually fail for some domains. Specifically, we produce an automorphism $\phi$ on a planar domain $\Omega$, such that the iterates of $\phi$ converge to a unique Euclidean boundary point, but do not converge to a unique Martin point in the Martin compactification of $\Omega$. We then extend this example to a family of examples in the second part of this work. We thus consider the Martin boundary for domains whose complement is contained in a strip and generalize results of Benedicks and Ancona.

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Additional Information

Pietro Poggi-Corradini
Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506
Address at time of publication: Department of Mathematics, East Hall, University of Michigan, Ann Arbor, Michigan 48109

Received by editor(s): April 6, 2001
Received by editor(s) in revised form: November 7, 2001
Published electronically: January 24, 2002
Additional Notes: The author was partially supported by NSF Grant DMS 97-06408. We thank Professor A. Ancona for very helpful conversations
Article copyright: © Copyright 2002 American Mathematical Society