preperiodic

Preperiodic is a term used in dynamical systems to describe points whose forward orbit under iteration of a function becomes periodic after finitely many steps. If f is a function from a set X to itself, a point x in X is preperiodic for f when there exist integers m ≥ 0 and p > 0 such that f^(m+p)(x) = f^m(x). In this case, the orbit of x enters a cycle of length p after m steps. Points with m = 0 are simply periodic.

The concept is closely related to, yet distinct from, periodic points. Every periodic point is preperiodic (with m = 0), while a preperiodic point is one whose orbit eventually falls onto a periodic cycle after a finite number of iterations. The preperiod tail consists of points that are not themselves in the cycle but map into it after several iterations.

A simple example arises in the complex map f(z) = z^2 on the Riemann sphere. The point z = 0 is periodic (fixed), while z = -1 is preperiodic because f(-1) = 1 and f(1) = 1. Similarly, z = i maps to -1 and then to 1, entering a fixed cycle and thus being preperiodic as well. Many maps have rich sets of preperiodic points, including those with intricate Julia or Fatou sets.

In arithmetic dynamics, preperiodic points are of particular interest because, for a fixed rational function of degree at least two over a number field, the set of preperiodic points is finite (Northcott’s theorem). This finiteness contrasts with the typically infinite or continuous dynamical structures seen over other fields.