Consider the function $f : x \mapsto \lfloor 100 \cos(x) \rfloor$ on the real numbers.
I would like to know whether the functions $f_n := \underbrace{f \circ ... \circ f}_{n \text{ times }}$ (with $n \geq 1$) are linearly independent over $\Bbb R$ (in the real vector space $\Bbb R^{\Bbb R}$).
The answer to my previous question shows that if a map $f : \Bbb R \to \Bbb R$ can be extended into a holomorphic map $h : \Bbb C \to \Bbb C$ which is periodic and its image is uncountable, then the iterates of $f$ are linearly independent over $\Bbb R$.
But in my case, $f$ is not even continuous, so I can not apply the same idea. How would you proceed, then?
In your example (if you study the integral part of $100\cos(x)$), the image of $\mathbb{R}$ by $f$ is a finite set, say $A_1$. Put $A_n=f_n(\mathbb{R})$. We have that $A_ {n+1}=f(A_n)$, and that $A_{n+1}\subset A_n$. As $A_1$ is a finite set, the intersection of all the $A_n$, say $A$, is also a finite not empty subset of $\mathbb{R}$, and we have that there exists $N$ such that $A_n=A$ for $n\geq N$. Now $f$ restricted to $A$ is an application from $A$ to $A$, let say this is $g_1$ and of course this is also the case for the iterates $ f_n$ restricted to $A$, say $g_n$. Now as $A$ is finite, the vector space of the functions from $A$ to $\mathbb{R}$ is finite dimensional, say of dimension $m$. Then there exist a non trivial linear combination of $g_1,\cdots, g_{m+1}$, ie $b_1g_1(y)+\cdots +b_{m+1}g_{m+1}(y)=0$ for all $y \in A$. But this show that we have $b_1f_{N+1}(x)+\cdots+b_{m+1}f_{m+1+N}(x)=0$ for all $x\in \mathbb{R}$.