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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?

shawn_halayka
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    It's a Julia set if the iterating function is a nonconstant complex rational function (that is $p(z)/q(z)$ with $p$ and $q$ complex polynomials and we may assume no root(s) in common with at least one having degree larger than $1$ (because iterated Moebius transformations have fairly simple structure)). You seem to be modeling this on the phrase "Julia polynomial" which I don't recognize and don't find in general use (by hitting my references and searching with Google). Are you sure your motivation for this name is standard/common/generic? – Eric Towers Aug 27 '21 at 04:24
  • Dear Eric, thank you very much for your comment. My motivation is to give proper attribution. I'm just not very familiar with the subject. – shawn_halayka Aug 27 '21 at 15:44
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    Mandelbrot polynomials https://link.springer.com/chapter/10.1007/978-1-4614-7621-4_13 – Claude Aug 28 '21 at 20:19
  • Thanks Claude, I appreciate it. – shawn_halayka Aug 28 '21 at 22:58

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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 $$K(P):=\{ z \in \mathbb C : P^n(z) \text{ is bounded } \}.$$

Albert
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