We introduce a map $\mathbf{x} \to \mathbf{x'}$, defined as (for example on $\mathbb{R}^3$):
$$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$
Note that $f,g,h$ are not all linear (or at least, I'm not interested in the all linear case).
Imagine there is a function (family of functions) such that $F(\mathbf{x})=F(\mathbf{x'})$, or in our case:
$$F(x',y',z')=F(x,y,z)$$
For any $x,y,z \in \mathbb{R}$ (or any other interval) and $x',y',z'$ determined by the transformation on $x,y,z.$
How to find out if such a function exists or not for certain $f,g,h$?
Is there a systematic way of searching for this function for arbitrary $f,g,h$? Or for some special cases?
A relaxed condition is also possible ($\alpha$ is a constant):
$$F(x',y',z')=\alpha F(x,y,z)$$
As an interesting example we can point out Carlson's integral:
$$F(x,y,z)=\frac{1}{2} \int_0^\infty \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}$$
$$F(x,y,z)=F \left( \frac{x+\lambda}{4}, \frac{y+\lambda}{4}, \frac{z+\lambda}{4} \right)$$
$$\lambda=\sqrt{xy}+\sqrt{yz}+\sqrt{zx}$$
Edit
Title changed due to Christian Blatter's comment
This is a very general question! Here is an example of the kind of statements you have in mind: If $$\phi:=(f,g,h):\quad\Omega\to{\mathbb R}^3,\qquad {\bf x}\mapsto{\bf x'}:=\phi({\bf x})$$ has a fixed point ${\bf p}\in\Omega$, and $\|d\phi({\bf p})\|<1$, then the point ${\bf p}$ is an attracting fixed point. In such a case there is a basin of attraction $\Omega'\subset\Omega$ such that $$\lim_{n\to\infty}\phi^{\circ n}({\bf x})={\bf p}$$ for all ${\bf x}\in\Omega'$. It follows that for a continuous $\phi$-invariant $F$ one has $F({\bf x})\equiv F({\bf p})$ on $\Omega'$.