Period Lengths for Iterated Functions. (Preliminary Version)

Let Ωn be the n -element set consisting of functions that have [n] as both domain and codomain. Since Ωn is finite, it is clear by the pigeonhole principle that, for any f ∈ Ωn, the sequence of compositional iterates f, f , f , f (4) . . . must eventually repeat. Let T(f) be the period of this eventually periodic sequence of functions, i.e. the least positive integer T such that, for all m ≥ n, f (m+T ) = f . A closely related number B(f) = the product of the lengths of the cycles of f , has previously been used as an approximation for T. This paper proves that the average values of these two quantities are quite different. The expected value of T is 1 nn ∑ f∈Ωn T(f) = exp ( k0 3 √ n log n ( 1 + o(1) )) , where k0 is a complicated but explicitly defined constant that is approximately 3.36. The expected value of B is much larger: 1 nn ∑ f∈Ωn B(f) = exp ( 3 2 3 √ n(1 + o(1)) ) .