This question is motivated by [another question on math.stackexchange](https://math.stackexchange.com/questions/1052783/mandelbrot-and-julia-set).


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. 



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


