In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.

Given a mapping f from a set X into itself,

a point x in X is called periodic point if there exists an n>0 so that

where f_n is the nth iterate of f. The smallest positive integer n satisfying the above is called the prime period or least period of the point x. If every point in X is a periodic point with the same period n, then f is called periodic with period n (this is not to be confused with the notion of a periodic function).

If there exist distinct n and m such that

then x is called a preperiodic point. All periodic points are preperiodic.

If f is a diffeomorphism of a differentiable manifold, so that the derivative ${\displaystyle f_{n}^{\prime }}$ is defined, then one says that a periodic point is hyperbolic if

that it is attractive if