Consider $x>0$
Let
$$f(x)= 2x + x^5$$
$$g(x) = x + x^3$$
$$f(r(x))=r(f(x))=id(x)$$
$$g(u(x)) = u(g(x))=id(x)$$
Where $id(x)$ is the identity function mapping all values to itself.
Let $*^{[y]}$ denote the $y$ th iteration of $*$.
Now consider for $x>0$ :
$$ h(x) = \lim_{n \to +\infty} f^{[n]}( g(r^{[n]}(x)) ) $$
$$m(x) = max[ f^{[1]}( g(r^{[1]}(x)) ), f^{[2]}( g(r^{[2]}(x)) ), f^{[3]}( g(r^{[3]}(x)) ) , ...]$$
How are $h(x)$ and $m(x)$ related ? Are they equal for all $x>0$ ?
What are good closed forms or good closed form asymptotics for $h(x)$ and $m(x)$ when $x>0$ ?
