On fixed points of commuting analytic functions

Abstract

In this note we show that if /and g map the unit disc | z\ = 1 in the complex plane into itself in a continuous manner, if they are analytic in the open disc, and if they commute (f(g(z))=g(f(z)) lor all z), then they have a common fixed point (f(zo)=Zo = g(zQ)).More generally, any commuting family of such functions has a common fixed point.In 1954 Eldon Dyer raised the following question: If/ and g are two continuous functions that map a closed interval on the real line into itself and commute, must they have a common fixed point?The same question was raised independently by the author in 1955 and by Lester Dubins in 1956.A more general question was posed by Isbell [2] in 1957.These questions have not been answered.The author wishes to thank N. D. Kazarinoff for several helpful discussions of this material.Let G be a bounded connected open set in the plane and let Fa denote the family of all those analytic functions in G whose range is contained in G (f(G) EG).With the topology of uniform convergence in compact subsets, Fq becomes a metric space.The functions in Fq are uniformly bounded; hence [3, Chapter 2, §7] each sequence of elements of Fq contains a convergent subsequence (the limit function need not be in Fq).Note that F0 is a semigroup under composition of functions.The following lemma tells us that Fq is a topological semigroup, that is, the semigroup operation is jointly continuous.

Locations

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