On fixed points of commuting analytic functions

Abstract

In this note we show that if f and g map the unit disc I 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)) for all z), then they have a common fixed point (f(zo) =zo =g(zo)). More generally, any commuting family of such functions has a common fixed point. In 1954 Eldon Dyer raised the following question: If f 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 FG denote the family of all those analytic functions in G whose range is contained in G (f(G) CG). With the topology of uniform convergence in compact subsets, FG becomes a metric space. The functions in FG are uniformly bounded; hence [3, Chapter 2, ?7] each sequence of elements of FG contains a convergent subsequence (the limit function need not be in FG). Note that FG is a semigroup under composition of functions. The following lemma tells us that FG is a topological semigroup, that is, the semigroup operation is jointly continuous.