Is the Schröder Equation valid for higher dimensional iterated maps? The Schröder's functional equation is the eigenfunction equation for the composition operator given as:
$$
\psi \circ y (x) = s \cdot \psi(x) ~~~~~~~~~~~ (1)
$$
The interesting bit about it (at least for me) is related to the iteration of a function over a point. Suppose that $y,x \in \mathbb{R}$. If we denote $y(y(x)) = y \circ y(x) = y_2(x)$ we can extend the notation like so $y_n(x) = y \circ y_{n-1}(x)$ and, more generally $y_{n+m}(x) = y_n \circ y_m(x)$. It turns out that once a Schröder's conjugate $\psi(x)$ is found for $y(x)$, it is possible to compute the $n$-th composition of $y(x)$ straightforwardly like so:
$$
y_n(x) = \psi^{-1}(s^n \cdot \psi(x))
$$
By computing $y_n$ like this, not only the integer values of $n$ are well-defined, but also all the intermediate values, turning $n$ into a real parameter and the iterating function $y_n(x)$ into the Flow $y(n,x)$. Meaning that the entire orbit containing all iterates of $y$, including the fractional ones, is obtained. This equation is often solved for an arbitrary function $y(x)$ using the Carleman matrices to turn composition into multiplication. 
Handy as it is, I could only find discussions about the Schröder equation, and its solution strategy, in the one-dimensional domain. So my question is whether there is a more general version of the Schröder's Equation for iterated maps of greater dimensionality, which generalizes equation (1) like so:
$$
\vec \varphi \circ \vec u(\vec x) = s \cdot \vec \varphi(\vec x) ~~~ \forall \vec \varphi, \vec x, \vec u \in \mathbb{R}^n, s \in \mathbb{R}.
$$
And if there is, whether or not we have a quasi-Carleman method to solve it as well.
 A: Frankly, I have not seen many multivariate techniques for functional iteration. You might consider experimenting with powers, emulating Schroeder's technique. Take two variables, x and y, for simplicity.
Again for simplicity, consider diagonal vector functions, e.g.,
$$
\vec u (x,y)= (x^2,y^3) ,
$$
so 
$$
\vec u_n(x,y)=(x^{2^n},y^{3^n}).
$$
I guess 
$$
\vec \varphi (\vec u (x,y)) = \operatorname{diag}(2,3) ~~\vec \varphi (x,y)
$$
would do the trick for $\vec \varphi (x,y)=(\log x , \log y)$, so, by functional conjugacy,
$$
\vec u _n (x,y)= \vec \varphi^{-1}(\operatorname{diag}(2^n,3^n)~\vec \varphi(x,y) ).
$$
Of course, for a scalar Schroeder eigenvalue as you have, you need the diagonal eigenvalue matrix to be proportional to the identity, so both entries 2 or 3 in our case. I took the slight generalization here to relax this restriction, even though there is no essential coupling between the two modes, so their orbits simply do not influence each other. 
As for perturbative techniques around fixed points, given the suitable equation arrays, such are always available. But make no mistake, the above diagonal generalizations are trivial, i.e. they are simply addressing a multiplexing of the scalar equation of one variable, with innocuous parallel moves and no essential coupling. In point of fact, the renormalization group equation for multicoupling QFT systems is of this form, but people bypass this entire structure in favor of coupled ODE systems—not functional equations anymore. I can see you asking for more elaborate constructs next...
