This is a multi-part Question. Please chime in with any interesting insights in addition to Answers.
I have noticed some interesting properties of Mandelbrot series that lead to a different way to plot the M-Set to elucidates certain details and properties. The 17,723 points in the graphic below all lie exactly on the edge of the M-Set.
Definitions: $z_{n}\equiv z_{n-1}^{2}+c$, $\,\,z_{0}\equiv c $.
Define the quality of this series being “$p$-periodic at $c$ from step $m$” if $z_{m}(c)=z_{m+p}(c)$ and $m,p$ are the lowest non-zero integers for which this is true at $c$. I'm making up terminology here, so forgive me if this already has a name.
If $z_m$ is periodic at $c$ from step $0$, i.e., $z_{m}(c)=c$, then some finite region around $c$ is within the M-set. However...
LEMMA: If $z$ is $p$-periodic at $c$ with any $p$ from step $m>0$, then $c$ is on the edge of the M-set and $\frac{d}{dc}z_{n}(c)$ diverges as $n\rightarrow\infty$. This is easy to demonstrate, but I haven't found a proof.
Question: Is this true? Can it be proven? Is this a well known thing?
The M-set does not have edge segments, since its edge is infinitely finely detailed, but the lemma allows us to generate an arbitrarily large number of points that are exactly on the edge.
The Mathematica code below generates a set of polynomials $\{u_{mn}\}$ where $z$ is $(m-n)$-periodic at $c$ from $n$ iff $u_{mn}(c)\equiv 0$. The roots for $n>0$ (M-set edges) are then calculated numerically and plotted.
I break the $z$ polynomials down into $\{u_{mn}\}$ polynomials in order to reduce as much as possible the order of polynomials that need to be solved numerically. For example, to calculate the $c$'s that are 3-periodic from 5, we would find them within the roots of $z_{8}(c)-z_{5}(c)$ but that set will also contain all the roots of $\{z_{7}-z_{4},z_{6}-z_{3},z_{5}-z_{2},z_{4}-z_{1},z_{3}-z_{0},z_{6}-z_{5},z_{5}-z_{4},z_{4}-z_{3},z_{3}-z_{2},z_{2}-z_{1},z_{1}-z_{0}\}$ so clearly, strategic factoring of the $z$ polynomials into u's and eliminating redundant root-finding saves significant time.
Question: Could the $u$ polynomials be analytically factored down further?
Question: Are there other properties of M-set edge points that would allow us to find more without huge computational cost?
z[n_, c_] := If[n > 0, z[n - 1, c]^2 + c, c];
ord = 8;
(* Calculate u[m,n] up to m\[Equal]ord *)
Do[Do[
If[n > 0, t = Expand[z[m, c] - z[n, c]], t = Expand[z[m - 1, c]]];
p = m - n;
Do[Do[If[((i != m) || (j != n)) && (Mod[p, i - j] == 0),
While[(tt = PolynomialQuotientRemainder[t, u[i, j], c])[[2]] == 0, t = tt[[1]]]], {j, 0, Min[n, i - 1]}], {i, 1, m}];
u[m, n] = t, {n, 0, m - 1}], {m, 1, ord}];
Print["Polynomial orders : ", Table[Exponent[u[m, n], c], {m, 1, ord}, {n, 0, m - 1}] // MatrixForm];
(* Compile numerical roots of u[m,n>0], which are c's on the edge of the M-set *)
plotOrd = 8;
$MaxRootDegree = Max[$MaxRootDegree, 2^(plotOrd - 1)];
rts = {};
Do[
Do[
s[m, n] = Solve[u[m, n] == 0, c] // N;
rts = Append[rts, c /. s[m, n]], {n, 1, m - 1}], {m, 1, plotOrd}];
rts = Flatten[rts];
Print["Number of Plot points : ", Length[rts]];
Print[ListPlot[Transpose[{Re[rts], Im[rts]}],PlotStyle ->PointSize[Small]]];
These settings will produce a plot in few seconds. The jpeg above took a while and was generated with
ord=11;
plotOrd=11;
and
Print[ListPlot[Transpose[{Re[rts], Im[rts]}],PlotStyle ->PointSize[Tiny]]];
It's also interesting the plot the roots of one u[m,n] at a time and observe where they end up and how the shape of the M-set emerges a m increases--especially around the crevice near c=1/4.
This Question connects thematically with another question about properties of the M-set accessible without knowledge of topology.