periodic point

Let $f:X\to X$ be a function and $f^{n}$ its $n$ -th iteration. A point $x$ is called a periodic point of period $n$ of $f$ if it is a fixed point of $f^{n}$ . The least $n$ for which $x$ is a fixed point of $f^{n}$ is called prime period or least period.

If $f$ is a function $\mathbb{R}$ to $\mathbb{R}$ or $\mathbb{C}$ to $\mathbb{C}$ then a periodic point $x$ of prime period $n$ is called hyperbolic if $|(f^{n})^{\prime}(x)|\neq 1$ , attractive if $|(f^{n})^{\prime}(x)|<1$ and repelling if $|(f^{n})^{\prime}(x)|>1$ .

Title	periodic point
Canonical name	PeriodicPoint
Date of creation	2013-03-22 12:43:38
Last modified on	2013-03-22 12:43:38
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	14
Author	mathwizard (128)
Entry type	Definition
Classification	msc 26A18
Defines	hyperbolic periodic point
Defines	attractive periodic point
Defines	repelling periodic point
Defines	least period
Defines	prime period