Preperiodicity

Preperiodicity refers to a property of points under the iteration of a function, where the forward orbit of the point eventually settles into a periodic cycle. In a dynamical system consisting of a map f from a set X to itself, a point x is periodic if some iterate f^n(x) equals x for a positive n. A point is preperiodic if there exist nonnegative integers m and p with p > 0 such that f^{m+p}(x) = f^m(x). In other words, after m steps the orbit becomes periodic with period p.

The initial segment of the orbit, up to the first appearance of the cycle, is called the tail, and its length is the preperiod. The cycle consists of p distinct points: f^m(x), f^{m+1}(x), ..., f^{m+p-1}(x). A point with m = 0 is already periodic, while a point with m > 0 becomes part of a cycle only after several iterations.

Examples help clarify the concept. For the real map f(x) = x^2 − 1, the point 1 maps to 0, which then enters the 2-cycle 0 ↔ −1, so 1 is preperiodic with tail length 1 and period 2. The point 2 maps to 3, then to 8, and so on, with no finite repetition in the forward orbit, so it is not preperiodic in the usual sense.

In finite settings, every point is preperiodic because orbits must eventually repeat. In broader contexts, preperiodicity characterizes how far a point is from entering the long-term, repeating behavior of the system.