Functions and its powers

Question

Given a map $\pi: A \rightarrow B$ what is the definition of $\pi^n$ where $n$ is a positive integer? For example if $\pi(a)=b$ then is $\pi^n(a)=b^n$?

Ok so if $n=3$ then $\pi^3(a)=\pi(\pi(\pi(a)))$. Isn't it?

Answer 1

More like $\pi(a)=a^2$ then

$$\begin{equation} \pi^2(a)=\pi(\pi(a))=(a^2)^2=a^4 \end{equation}$$

Answer 2

In operator theory, the $n$-th power $T^n$ of $T$ really means $n$-times composite of $T$, which is a operator. Note that the range $R(T)$ of $T$ must be a subset of the domain $D(T)$ in order for $T^n$ to be well-defined.

$T^n(a) = T(a)^n$ may not be true even if $T$ is indeed a function $f: \mathbb{R} \rightarrow \mathbb{R}$. Consider $f(x) = x-1$. Then $f^2(2)=f(f(2))=f(1)=0$ and $f(2)^2 = 1^2 = 1$.