Periodic point

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In mathematics, a periodic point x is a point for which <math>f^n(x)=x</math>, where <math>f^n</math> is the nth iterate of some function. The smallest positive integer n satisfying the above is called the prime period or least period of the point x. If every point in the domain of f is a periodic point with the same period n, then f is a periodic function with period n.

Periodic points are common objects of study in topology and dynamical systems.

A period-one point is called a fixed point.

If f is a diffeomorphism of a differentiable manifold, so that the derivative <math>(f^n)^\prime</math> is defined, then one says that a periodic point is hyperbolic if

<math>|(f^n)^\prime|\ne 1,</math>

and that it is attractive if

<math>|(f^n)^\prime|< 1</math>

and it is repelling if

<math>|(f^n)^\prime|> 1.</math>

If the dimension of the stable manifold of a periodic point or fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle or saddle point.