TITLE: Dynamics in one matrix variable
QUESTION [12 upvotes]: Are dynamical systems
$$X \mapsto F(X)$$
studied where $X \in \mathrm{M}_n$, $\mathrm{M}_n:=\mathrm{Mat}(n,\mathbb{C})$ or $\mathrm{Mat}(n,\mathbb{R})$, and $F$ is a (properly defined noncommutative) polynomial function?
For example, I was wondering if anybody has studied "Julia sets" and "Mandelbrot set" for mappings of the form
$$X \mapsto X^2 + C$$
with $C\in\mathrm{M}_n$.
One could presumably ask the same questions in the case of "one Clifford variable", i.e. for $X\in$ the Clifford algebra $\mathcal{C}\ell(V,Q)$ where $V$ is a (real or complex) finite dimentional vector space and $Q$ a quadratic form on $V$. In this case, for example, $X^2+C$ would reduce to $Q(X)\cdot 1+C$ for all $X \in V$.

REPLY [3 votes]: To answer your first, but not the rest of the questions, I thought about fractional linear matrix transformations (or at least their denominators), that is, densely defined maps on $n \times n$ matrices, of the form $X \mapsto (I + AXB)^{-1}$ (Fixed points of two-sided fractional matrix transformations, Fixed point theory and applications, 2007, doi:10.1155/2007/41930), but only with respect to fixed points. At least for the purpose of describing the fixed points, these can be rewritten in the form of matrix Riccati equations, and are closely connected to transformations of the type $X \mapsto X^2 + C$. But this probably isn't what you're looking for.