Iterated Function - Fractional Iterates and Flows, and Negative Iterates

Fractional Iterates and Flows, and Negative Iterates

In some instances, fractional iteration of a function can be defined: for instance, a half iterate of a function f is a function g such that g(g(x)) = f(x). This function g(x) can be written using the index notation as f ½(x). Similarly, f ⅓(x) is the function defined such that f1/3(f1/3(f1/3(x))) = f(x), while f ⅔(x) may be defined equal to f ⅓(f ⅓(x)), and so forth, all based on the principle, mentioned earlier, that f m ∘ f n = f m + n. This idea can be generalized so that the iteration count n becomes a continuous parameter, a sort of continuous "time" of a continuous orbit.

In such cases, one refers to the system as a flow, specified by Schröder's equation. (cf. Section on conjugacy below.)

Negative iterates correspond to function inverses and their compositions. For example, f −1(x) is the normal inverse of f, while f −2(x) is the inverse composed with itself (i.e. f −2(x) = f −1(f −1(x)).) Fractional negative iterates are defined analogously to fractional positive ones; for example, f −1/2(x) is defined such that f − ½(f −½(x)) = f −1(x), or, equivalently, such that f −½(f ½(x)) = f 0(x) = x.