On fixed points of commuting functions
Haskell Cohen · Proceedings of the American Mathematical Society · 1964
There is a rather well-known conjecture that if f and g are continuous functions on [0, 1] to itself which commute (i.e., f(g(x)) =g(f(x))), then they have a common fixed point. The conjecture is apparently due independently to Eldon Dyer and Allen Shields, and has been generalized by J. R. Isbell [2]. The conjecture is easily verified for polynomials f and g by referring to some work of J. F. Ritt [3] who showed that the only commuting polynomials, aside from some trivial cases are the Tchebycheff polynomials all of which have a common fixed point. This result is stated more explicitly by Block and Thielman [I]. The author has noted that certain functions with broken line graphs, e.g.,