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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Angular derivatives at boundary fixed points
for self-maps of the disk


Author: Pietro Poggi-Corradini
Journal: Proc. Amer. Math. Soc. 126 (1998), 1697-1708
MSC (1991): Primary 30D05, 30D55, 30C35, 47B38, 58F23
MathSciNet review: 1443404
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Abstract: Let $\phi$ be a one-to-one analytic function of the unit disk $\mathbb{D} $ into itself, with $\phi(0)=0$. The origin is an attracting fixed point for $\phi$, if $\phi$ is not a rotation. In addition, there can be fixed points on $\partial \mathbb{D} $ where $\phi$ has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of $\phi$ is a one-to-one analytic function $\sigma $ defined on $\mathbb{D} $ such that $\phi= \sigma ^{-1}(\lambda \sigma )$, where $\lambda =\phi^\prime(0)$. If $\phi _K$ is the first iterate of $\phi$ that does have b.r.f.p., we compute the Hardy number of $\sigma $, $h(\sigma )=\sup\{p>0:\ \sigma \in H^p(\mathbb{D} )\}$, in terms of the smallest angular derivative of $\phi _K$ at its b.r.f.p.. In the case when no iterate of $\phi$ has b.r.f.p., then $\sigma \in \bigcap _{p<\infty}H^p $, and vice versa. This has applications to composition operators, since $\sigma $ is a formal eigenfunction of the operator $C_\phi (f)=f\circ\phi$. When $C_\phi $ acts on $H^2(\mathbb{D} )$, by a result of C. Cowen and B. MacCluer, the spectrum of $C_\phi $ is determined by $\lambda $ and the essential spectral radius of $C_\phi $, $r_e(C_\phi )$. Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, $r_e(C_\phi )$ can be computed in terms of $h(\sigma )$. Hence, our result implies that the spectrum of $C_\phi $ is determined by the derivative of $\phi$ at the fixed point $0\in \mathbb{D} $ and the angular derivatives at b.r.f.p. of $\phi$ or some iterate of $\phi$.


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Additional Information

Pietro Poggi-Corradini
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email: pp2n@virginia.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04303-2
PII: S 0002-9939(98)04303-2
Received by editor(s): January 18, 1996
Received by editor(s) in revised form: November 15, 1996
Additional Notes: The author was supported by the University of Washington Math. Department while at MSRI, Berkeley, in the Fall of 1995. He also wishes to thank Professor D. Marshall for his help and advice.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society