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.

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 ${\displaystyle f_{n}}$ 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 ${\displaystyle f_{n}^{\prime }}$ is defined, then one says that a periodic point is hyperbolic if

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