# periodic point

Let $f:X\to X$ be a function and $f^{n}$ its $n$-th iteration. A point $x$ is called a 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 PeriodicPoint 2013-03-22 12:43:38 2013-03-22 12:43:38 mathwizard (128) mathwizard (128) 14 mathwizard (128) Definition msc 26A18 hyperbolic periodic point attractive periodic point repelling periodic point least period prime period