Mandelbrot fractal by matrix powers? Would such a construction let us analyze it with linear algebra?

Question

The famous Mandelbrot fractal can be calculated by starting $z_0=0$ and then iterating

$$z_{n+1} = {z_n}^2+c$$

and seeing which $c$ makes $z$ run off to complex infinity ( and if so, how fast ).

Can we represent this with a matrix? If we can, would that let us analyze it's behavior with linear algebra?

Answer 1

Have you heard about IFS (https://en.wikipedia.org/wiki/Iterated_function_system) (http://www.hiddendimension.com/fractalmath/ifs_fractals_main.html)? They use linear algebra and generate fractals in the same "spirit" as Mandelbrot fractals (https://mathoverflow.net/q/199286).

Answer 2

That's depend on what you mean by linear algebra. The equation is certainly not linear and it does not become linear because you use matrices (from linear algebra). After all the matrices are not more "linear" than the real numbers (as the real numbers and $1\times 1$ matrices are basically the same).

If you extend the recursive formula to matrices then one can easily see that it's required that the matrices are all square and of the same dimension. I don't think there's much understanding to be gained by this either, for example if you (can) diagonalize $c$ you see that the recursion formula only decomposes to a set of recursion formulas of the original kind (you'll only get back to square one again).