Notation for Functional Iteration

Question

In my research, I often use functional iteration, and I usually denote $$(f\circ f\circ f\circ\dots\circ f)(x)$$ by writing $$f^n(x)$$ where $f$ was composed with itself $n$ times. However, this notation can be confusing at times, especially when dealing with the trigonometric functions. Does anyone know of any elegant ways of expressing iterated function composition that do not conflict with preexisting notations?

Answer 1

From Wikipedia:

To avoid ambiguity, some mathematicians choose to write $f^{\circ n}$ for the $n$-th iterate of the function $f$.

Answer 2

Hint: The paper Notations for Iteration of Functions, Iteral by Valerii Salov might be interesting. He proposes a new notation and gives in section 3 a short survey about existing notation.

He refers in this section besides some others to

• Iterative Functional Equations by M. Kuczma, B. Choczewski, R. Ger stating that composition $\circ$ of functions is the only inner operation which be defined in the family $\mathcal{F}(x)$ of self-mappings for a set $X$. The system $(\mathcal{F}(X),\circ)$ is with $\mathrm{id}_X$ a non-commutative monoid with iterates defined as the powers of $n$ of an element $f\in\mathcal{F}(X)$, where $n$ is a non-negative integer: \begin{align*} n\in\mathbb{N},f^0=\mathrm{id}_X,f^{n+1}=f\circ f^n \end{align*}

another notation was introduced by D. Knuth in

• The Art of Computer Programming, Vol. 2 Seminumerical algorithms. There he chooses square parentheses to reduce ambiguity with powers of multiplication. \begin{align*} f^{[n]}(x)=f(f^{[n-1]}(x)) \end{align*}