Dynamics in one matrix variable

Question

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 $X \in V$.

Answer 1

A few elementary (and probably not very useful) thoughts:

You can always write these simply as polynomial maps of affine space, and then they are special cases of general theorems about polynomial maps of affine spaces. So presumably what you want to know is if you can use the matrix formulation to glean additional information. Have you looked at the $n=2$ case. Writing $X=\left(\begin{smallmatrix} x&y\\ z&w\\ \end{smallmatrix}\right)$ and $C=\left(\begin{smallmatrix} a&b\\ c&d\\ \end{smallmatrix}\right)$, the map $X\mapsto X^2+C$ is simply the map $$ F : \mathbb{A}^4\to\mathbb{A}^4,\qquad (x,y,z,w) \mapsto (x^2+yz+a, xy+wy+b, xz+wz+c, w^2+yz+d). $$ Homogenizing gives $\bar F : \mathbb{P}^4\to\mathbb{P}^4$, $$ \bar F(x:y:z:w:t) = (x^2+yz+at^2: xy+wy+bt^2: xz+wz+ct^2: w^2+yz+dt^2 : t^2) $$ The indeterminacy locus on the hyperplane at infinity is the rational curve parametrized by (if I've computed correctly) $$ \{ (uv, u^2,-v^2, -uv, 0) : u,v\in\mathbb{C} \}. $$

What do you mean by "the" Mandelbrot set in this setting? I guess one could define it to be the points $C$ such that the orbits of the critical points are all bounded. Of course, the space of $C$ is not 4 dimensional, since you can always conjugate by an arbitrary invertible matrix. If you're working over $\mathbb C$, then you may assume that $C$ is in Jordan normal form, so the moduli space consists of two components, namely diagonal $C$ and $C$ consisting of a single non-semisimple Jordan block. You might consider each of these in turn.

Answer 2

Julia sets and Mandelbrot sets for polynomial functions on the Quaternions $\mathbb{H}$ are studied, see for instance http://www.evl.uic.edu/hypercomplex/html/book/book.pdf. $\mathbb{H}$ may be described as a subring of $M(2,\mathbb{C})$ or $M(4,\mathbb{R})$. Also Julia sets for polynomial skew products in $\mathbb{C}^2$ are studied, see for instance http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.642.

Ok, this answers Your question only in a special setting, and I have not heared about a general theory.

Answer 3

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.