Doing composition of functions with my students and was wondering if there was a large composition operator similar to Sigma and Pi? What I'm thinking is composing a function n times... $$(f\circ f\circ ... \circ f)(x)=f(f(...(f(x))...))=\circ_{i=1}^n{f}$$.
Not an overly important question, but i would love to maybe illustrate this to some of the more exceptional kids in my class who are aware of operators such as $\Sigma$ and $\Pi$.
If you want to get really deep, consider chapter 2.4 of Cowen & MacCluer's book, Composition Operators on Spaces of Analytic Functions here. Actually, this iteration of composition is a primary tool! Their setting is usually $H^2$, the Hardy space of analytic functions on the disk (roughly the functions which are square integrable on the unit circle, and have a taylor series on the disk). Fix a function, $\phi$ then map a function $f$ in $H^2$ to $f\circ \phi$ (also in $H^2$!). To study this linear map on $H^2$ (it's just a matrix if you like linear algebra), iterating it it very important!