I'm quite naive in math, and this question is to ask suggestions in how to pose a problem or to know if I'm just saying nonsense.
I have a map $\phi:\mathbb R^n\times\mathbb R^m\to\mathbb R^m$, for some $n,m\in\mathbb N_{>0}$, that is as much regular as needed. Given a $x\in\mathbb R^n$, I can define the mapping $\phi_x:\mathbb R^m\to\mathbb R^m$ as $\phi_x(\cdot) = \phi(x,\cdot)$.
I was trying to falsify a condition, that I will call $C$. Thus I assumed $C$ to hold, and at a certain point I found that $C$ implies the following $$ c=\Big(\phi_a\circ\phi_a\circ\phi_a\circ\cdots\Big)(b)\qquad\qquad(1) $$ with $a\in\mathbb R^n$ and $b,c\in\mathbb R^m$ some given points and with the object "$\phi_a\circ\phi_a\circ\phi_a\circ\cdots$" that is to be interpreted as the infinite compositions of the map $\phi_a$.
Now, I think that I could neither write (1) as far as I don't know that "$\phi_a\circ\phi_a\circ\phi_a\circ\cdots$" makes sense, and that's the point. I actually do not want (1) to make sense, since if (1) does not, that would imply that $C$ cannot hold (by the way, am I correct in saying so?). Thus essentially my preliminary strategy is to find sufficient conditions on $\phi_a$ under which (1) does not make sense.
As I want to study the "non-existence" (just to say) properties of "$\phi_a\circ\phi_a\circ\phi_a\circ\cdots$", it seems to me fair to start working with something I can define. For instance, given $k\in\mathbb N$, by letting $$ \Phi_a^k := \underbrace{\phi_a\circ\cdots\circ \phi_a}_{k\;times} $$ I can try to study something like the existence of $\lim_{k\to\infty}\Phi_a^k$, with $\lim$ that I think is to be intended as a pontwise limit.
I found many references on sufficient conditions for the existence of limits of the kind of $\lim_{k\to\infty}\Phi_a^k$, however I did not find anything useful on the non-existence of such objects (for instance conditions under which the sequence of $\Phi_a^k$ does not converge to any function).
I'm also interested in results saying that if $\lim_{k\to\infty}\Phi^k_a$ exists or if (1) holds then necessarily $\phi_a$ must have some given properties ( thus necessary conditions rather than sufficient) or something characterising the values of $(b,c)$ for which (1) holds, or don't).
Do you have any hint?
thanks.