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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Commuting analytic functions without fixed points

Author: Donald F. Behan
Journal: Proc. Amer. Math. Soc. 37 (1973), 114-120
MSC: Primary 30A20
MathSciNet review: 0308378
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Abstract: Let A be the set of nonidentity analytic functions which map the open unit disk into itself. Wolff has shown that the iterates of $ f \in A$ converge uniformly on compact sets to a constant $ T(f)$, unless f is an elliptic conformal automorphism of the disk. This paper presents a proof that if f and g are in A and commute under composition, and if f is not a hyperbolic conformal automorphism of the disk, then $ T(f) = T(g)$. This extends, in a sense, a result of Shields. The proof involves the so-called angular derivative of a function in A at a boundary point of the disk.

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Keywords: Commuting under composition, iteration, fixed point, angular derivative, Julia lemma, chain rule, Lindelöf theorem
Article copyright: © Copyright 1973 American Mathematical Society