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

### Examples

## Dynamical system

### Properties

## See also

Given an endomorphism f on a set X

- $f:\; X\; to\; X$

- $f^n(x)\; =\; x$

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

- $|(f^n)^prime|ne\; 1,$

and that it is attractive if

- $|(f^n)^prime|<\; 1$

and it is repelling if

- $|(f^n)^prime|>\; 1.$

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.

- A period-one point is called a fixed point.

Given a real global dynamical system (R, X, Φ) with X the phase space and Φ the evolution function,

- $Phi:\; mathbb\{R\}\; times\; X\; to\; X$

- $Phi(t,\; x)\; =\; x,$

- Given a periodic point x with period t, then $Phi(s,\; x)\; =\; Phi(s\; +\; t,\; x),$ for all s in R
- Given a periodic point x then all points on the orbit $gamma\_x$ through x are periodic with the same prime period.

- Limit cycle
- Limit set
- Stable set
- Sharkovsky's theorem
- Stationary point
- Periodic points of complex quadratic mappings

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Last updated on Tuesday July 17, 2007 at 04:22:13 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday July 17, 2007 at 04:22:13 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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