Commuting functions with no common fixed point

Abstract

Let/and g be continuous functions mapping the unit interval / into itself which commute under functional composition, that is, f(g(x)) = g(f(x)) for all x in /. In 1954 Eldon Dyer asked whether/and g must always have a common fixed point, meaning a point z in / for which f(z) = z=g(z). A. L. Shields posed the same question independently in 1955, as did Lester Dubins in 1956. The problem first appears in the literature in The purpose of this paper is to answer Dyer's question in the negative by the construction of a pair of commuting functions which have no fixed point in common.