Periodic point

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. 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 an endomorphism f on a set X a point x in X is called periodic point if there exists an n so that where f n {displaystyle 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. 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 f n ′ {displaystyle f_{n}^{prime }} is defined, then one says that a periodic point is hyperbolic if that it is attractive if and it is repelling if