# Definition:Power of Mapping

## Definition

Let $f: S \to S$ be a mapping from $S$ to itself.

Because the domain of $f$ is equal to the codomain of $f$ (both are $S$), the composite mapping $f \circ f$ is defined.

We define the $n$th power of $f$ as:

$\forall n \in \N: f^n = \begin{cases} I_S & : n = 0 \\ f \circ f^{n - 1} & : n > 0 \end{cases}$

where $I_S$ is the identity mapping.