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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Iteration and the solution of functional equations for functions analytic in the unit disk


Author: Carl C. Cowen
Journal: Trans. Amer. Math. Soc. 265 (1981), 69-95
MSC: Primary 30D05; Secondary 39B05, 60J99
DOI: https://doi.org/10.1090/S0002-9947-1981-0607108-9
MathSciNet review: 607108
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Abstract: This paper considers the classical functional equations of Schroeder $f \circ \varphi = \lambda f$, and Abel $f \circ \varphi = f + 1$, and related problems of fractional iteration where $\varphi$ is an analytic mapping of the open unit disk into itself. The main theorem states that under very general conditions there is a linear fractional transformation $\Phi$ and a function $\sigma$ analytic in the disk such that $\Phi \circ \sigma = \sigma \circ \varphi$ and that, with suitable normalization, $\Phi$ and $\sigma$ are unique. In particular, the hypotheses are satisfied if $\varphi$ is a probability generating function that does not have a double zero at $0$. This intertwining relates solutions of functional equations for $\varphi$ to solutions of the corresponding equations for $\Phi$. For example, it follows that if $\varphi$ has no fixed points in the open disk, then the solution space of $f \circ \varphi = \lambda f$ is infinite dimensional for every nonzero $\lambda$. Although the discrete semigroup of iterates of $\varphi$ usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself, we find that for each $z$ in the disk, all sufficiently large fractional iterates of $\varphi$ can be defined at $z$. This enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of $\varphi$. If the iterates of $\varphi$ can be embedded in a continuous semigroup, we show that the semigroup must come from the corresponding semigroup for $\Phi$, and thus be real analytic in $t$. The proof of the main theorem is not based on the well known limit technique introduced by Koenigs (1884) but rather on the construction of a Riemann surface on which an extension of $\varphi$ is a bijection. Much work is devoted to relating characteristics of $\varphi$ to the particular linear fractional transformation constructed in the theorem.


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Keywords: Functional equation, iteration, analytic function, semigroup, infinitesimal generator, Galton-Watson process
Article copyright: © Copyright 1981 American Mathematical Society