I am trying to learn more about infinite application of functions and functionals. My background is in quantum chemistry, so please forgive some of my notation and terminology.
Motivation
In analysis, we speak of the infinite differentiability of a function in order to determine some properties of the smoothness and analyticity of a function:
$f^{\infty}(x)=\frac{d}{dx}\frac{d}{dx}\cdots\frac{d}{dx}f(x)=F[F[\cdots F[f(x)]]]$
where $F[f(x)]=\frac{d}{dx}f(x)$ and $F = \frac{d}{dx}$.
In this case I believe that $F$ is the derivative functional, and we are applying it an infinite number of times to the function $f(x)$. In limit notation we might write:
$\lim_{n\rightarrow\infty}F^n[f(x)]=f^\infty(x)$.
I am curious how we can extend this notion to other functions and functionals.
Specific Question
Carrying this notion forward, consider the functional of a bounded function like sine: $F[f(x)] = \sin(f(x))$. Now apply the functional an infinite number of times. In the limit notation above, $\lim_{n\rightarrow\infty}F^n[f(x)]=\sin(\sin(\cdots\sin(f(x))))$. To simplify the problem, set $f(x)=x$. What is the limit of applying the sine function an infinite number of times to some number between $0$ and $2\pi$? For example, $\lim_{n\rightarrow\infty}\sin^n(\frac{\pi}{2})=\,?$
Further Questions
- For a general functional applied an infinite number of times to a general function, how might I go about proving properties like convergence/divergence and oscillation behavior? Harking back to calculus, I am trying to generalize the various results of convergence and divergence tests to the application of functions.
- Is there a specific term in analysis for applying a function or functional an infinite number of times? I am at a loss right now for any information about this topic.