Is there literature available on iterated function systems of the form $f^n = (g f^{n - 1}, g f^{n - 2}, \ldots)$?

Question

This question is motivated by another question on math.stackexchange.

From a function $g:X^k\to X$ it is possible to define an iterated function system on $X^k$ with the function $f:X^k\to X^k$ defined by

$$f(x) = (g(x), x_1, x_2, \ldots, x_{k - 1}).$$

Iterates of $f$ are given by

$$f^n = f f^{n - 1} = (g f^{n - 1}, f^{n - 1}_1, \ldots, f^{n - 1}_{k - 1})$$

and when $n > k$ an exercise in tedious definition chasing yields

$$f^n = (g f^{n - 1}, g f^{n - 2}, \ldots, g f^{n - k}).$$

The linked question describes such a system where, $X = B(z, r)$, an open ball in $\mathbb{C}$, $k = 2$, and $g(x) = (\alpha + x_1)/(1 + x_2)$.

My question is:

Is there literature available describing these types of iterated function systems?

I wonder if these systems have a name. I have not had much luck trying to describe them to google.

In the special case where $k = 1$ these are just "normal" iterated function systems and there is a plethora of literature describing techniques for analyzing their dynamics; however it is not clear to me which tools I can justifiably bring to bear when the next point in the iteration depends on the current point and several previous points.

Answer 1

(More an answomment). I'd say your iteration on $X^k$ is what one gets by a standard reduction writing a $k$-order recurrence on $X$, $$x_{n}=g(x_{n-1},x_{n-2},\dots,x_{n-k}),\qquad x_i\in X$$ in form of a first-order recurrence on $X^k$, $$y^{n+1}=f(y^n),\qquad y^n=(y^n_1,\dots,y^n_k)\in X^k$$ putting $y^n_i:=x_{n-i}$. (By the analog reduction, one writes a $k$-order scalar ODE as a first order system on $\mathbb{R}^k$). Since $X$ in your setting seems to be quite a general object, your iteration $y\mapsto f(y)$ is actually a more special rather than more general iteration, and the question is, what implications has the special form of $f$ on the dynamics. To start with, for instance, if $g$ is a linear form on $X:=\mathbb{C}$, the corresponding matrix $f$ has all eigenvalues of geometric multiplicity $1$. I guess you may like to refer to the theory of higher order recurrences to check if some result may be of interest for you when traslated in your setting.