Dedicated to John Riordan on the Occasion of His 75th Birthday
نویسنده
چکیده
Baxter permutations apparently first arose in attempts to prove the “commuting function” conjecture of Dyer (see [I]), namely, if f and g are continuous functions mapping [0, l] into [0, l] which commute under composition, then they have a common fixed point. Although numerous partial results were obtained for the conjecture (e.g., see [l, 3, 7, IO]), it was ultimately shown in 1967 to be false by Boyce [5] and independently, by Hunecke [8]. However, it has recently been pointed out by Boyce [6] that Baxter permutations are of more general significance in analysis than had previously been realized. This comes about as follows. For a continuous function h : [0, l] -+ [0, 11, let [/z] = (x : h(x) = x) denote the set of fixed points of h and let [h]* C [h] denote the set of crossing points of h, i.e., 01 E [h*] if and only if (Y is a limit point of both {x : h(x) < x} and (x : h(x) > x} (if 01 = 0 then only the first condition must hold; if (Y = 1 then only the second must hold). For continuous functions f,g: [0, l]+[O, l],ifaE[gof]then
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