Iterated function

In mathematics, iterated functions are the objects of study in fractals and dynamical systems. The formal definition of an iterated function on a set X follows:

Let X be a set and

$f:X\rightarrow X$

be a mapping. Define the n'--th iterate

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f^n}

of the map by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f^0=\textrm{id}_X}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \textrm{id}_X} is the identity function on X, and

$f^{n+1}=f\circ f^{n}$.

Examples

Famous iterated functions include the Mandelbrot set and Iterated function systems.