Equivalence of exponent in functions?

Question

Is there any way of indicating that a function is supposed to be applied a certain number of times? (In such a way that you repeatedly take the output and put it back into the function.)

Let's say for example you have a function $a(x,y)$ which is defined in such a way that $a(x,1) = b(x)$, $a(x,2) = b(b(x))$, $a(x,3) = b(b(b(x)))$ and so forth... could you then simply use exponentiation and write $a(x,y) = b(x)^y$?

Answer 1

According to the Wikipedia entry iterated function, the $n$-th iterate of $f$ can be written as $f^n$, as long as you define it properly: this makes sense, if you consider the monoid of functions under composition.

Now, still quoting Wikipedia:

Because the notation $f^n$ may refer to both iteration (composition) of the function f or exponentiation of the function f (the latter is commonly used in trigonometry), some mathematicians choose to write $f^{\circ n}$ for the $n$-th iterate of the function $f$.

Finally, I fully agree with Toby Mak that $f^{(n)}$ is also of common use.

Conclusion. Whatever notation you choose, start by giving a precise definition to avoid any ambiguity. This definition can be very short, like "let $f^n$ denote the $n$-th iterate of the function $f$", but is mandatory.