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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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On the failure of a generalized Denjoy-Wolff theorem
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by Pietro Poggi-Corradini PDF
Conform. Geom. Dyn. 6 (2002), 13-32 Request permission

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
  • Email: pietro@math.ksu.edu, pietropc@umich.edu
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 6 (2002), 13-32
  • MSC (2000): Primary 30D05, 31A05
  • DOI: https://doi.org/10.1090/S1088-4173-02-00075-9
  • MathSciNet review: 1882086