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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
DOI: https://doi.org/10.1090/S1088-4173-02-00075-9
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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  • [An79] A. Ancona, Une propriété de la compactification de Martin d'un domaine euclidien, Ann. Inst. Fourier Grenoble, 29, 4 (1979), 71-90. MR 81f:31013
  • [An84] A. Ancona, Régularité d'accès des bouts, et frontière de Martin d'un domaine euclidien. (French) [Regularity of attainability of ends and Martin boundary of a Euclidean domain] J. Math. Pures Appl. (9) 63 (1984), no. 2, 215-260. MR 86f:31005
  • [AG00] D. Armitage and S. Gardiner, Classical potential theory. Springer, 2000. MR 2001m:31001
  • [ADP-C] V. Azarin, D. Drasin, P. Poggi-Corradini, A generalization of $\rho $-trigonometrically convex functions and positive harmonic functions in $T$-invariant domains, preprint.
  • [Ba95] R. Bass, Probabilistic techniques in analysis, Springer-Verlag, New York, 1995. MR 96e:60001
  • [Be80] M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in $\mathbb{R}^{n}$, Ark. Mat. 18 (1980), no. 1, 53-72. MR 82h:31004
  • [Ca82] L. Carleson, Estimates of harmonic measures. Ann. Acad. Sci. Fenn. Math. 7 (1982), no. 1, 25-32 MR 84i:30032
  • [Ga89] S. Gardiner, Minimal harmonic functions on Denjoy domains, Proc. Amer. Math. Soc. 107 (1989), no. 4, 963-970. MR 90c:31013
  • [He41] M. Heins, On the iteration of functions which are analytic and single-valued in a given multiply-connected region, American Journal of Mathematics, Vol. 63, No. 2. (Apr., 1941), 461-480. MR 2:275a
  • [He88] M. Heins, A theorem of Wolff-Denjoy type, in Complex Analysis, Birkhäuser, (1988), 81-86. MR 90d:30077
  • [Kje51] B. Kjellberg, On the growth of minimal positive harmonic functions in a plane region, Ark. Mat. 1 (1951), 347-351. MR 12:410f
  • [Ko88] P. Koosis, The logarithmic integral. I. Cambridge Studies in Advanced Mathematics, 12. Cambridge University Press, 1988. MR 90a:30097
  • [So94] M. Sodin, An elementary proof of Benedicks' and Carleson's estimates of harmonic measure of linear sets. Proc. Amer. Math. Soc. 121 (1994), no. 4, 1079-1085. MR 94j:30022

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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

DOI: https://doi.org/10.1090/S1088-4173-02-00075-9
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

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