Here is a plot of 17,723 Misiurewicz Points. The code below generates a set of polynomials u[m,n], the roots of which have periodicity (m-n) starting at iteration n. I stopped at 17,723 points because to get more by this method I had to generate and solve $2^{11}$-order polynomials. In other words, I hit the practical limit of my method.


So my question is: Is there any way I can find more Misiurewicz points without having to solve giant polynomials?

This relates to another Question about the right-most real Misiurewicz Point.

Mathematica code:

z[n_, c_] := If[n > 0, z[n - 1, c]^2 + c, c];
ord = 8;
(* Calculate u[m,n] up to m==ord *)
   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 = {};
   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



Print[ListPlot[Transpose[{Re[rts], Im[rts]}],PlotStyle ->PointSize[Tiny]]];
  • $\begingroup$ The parabolic and Misiurewicz points can be parametrized by the rational numbers between 0 and 1 using the idea of an external ray. The Wikipedia page provides a serviceable introduction, this paper provides a modern account, and Milnor's book is excellent as well. External rays are not hard to compute and you can use one to get close to a Misiurewicz point. I could provide more details over on Mathematica.se, if you like. $\endgroup$ Feb 6 '16 at 23:31
  • $\begingroup$ @MarkMcClure Thanks, that would be great. But I think it would be kosher to post MMa code here, since it's the answer to a math question. $\endgroup$ Feb 6 '16 at 23:47
  • $\begingroup$ @MarkMcClure paper link has rotted. Archive copy: web.archive.org/web/20170519212111/http://… Rational Parameter Rays Of The Mandelbrot Set Dierk Schleicher $\endgroup$
    – Claude
    May 23 '19 at 14:06
  • $\begingroup$ @MarkMcClure book link has rotted. Dynamics in One Complex Variable: Introductory Lectures John Milnor (2nd Ed, 2000) $\endgroup$
    – Claude
    May 23 '19 at 14:14

Using numerical methods, like:

  1. Misiurewicz domains
  2. Newton's method for Misiurewicz points

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