Commuting analytic functions without fixed points

Behan, Donald F.

doi:10.1090/S0002-9939-1973-0308378-8

Commuting analytic functions without fixed points
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by Donald F. Behan

Proc. Amer. Math. Soc. 37 (1973), 114-120

DOI: https://doi.org/10.1090/S0002-9939-1973-0308378-8

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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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Sur l’itération des fonctions holomorphes dans un demi-plan

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Sur l’itération des fonctions bornées