Polynomials of a particular form

Question

I have an iterative polynomial in fractal geometry, namely $Z = Z^2 + C$. What is the name of the polynomial of the more general form $Z = Z^\beta + Z^\gamma + ... + C$?

I am calling them Julia-Fatou-Mandelbrot polynomials. Is that incorrect?

Answer 1

The notation $Z=Z^2+C$ or the term "iterative polynomials" are non-standard and misleading.

One usually just give a name to the polynomial $z \mapsto z^2+c$ (for instance, $f_c$) and then one can talk about the Julia set of the polynomial $f_c$, usually denoted by $J(f_c)$. As you know, the definition of this set involves the iterates $f_c^n:=f_c \circ \ldots \circ f_c$.

More generally, if $P$ is any polynomial (usually one also requires degree at least two to rule out trivial cases), you define the Julia set of $P$ (denoted by $J(P)$) in the same way, for instance as the boundary of the set $$K(P):=\{ z \in \mathbb C : P^n(z) \text{ is bounded } \}.$$