"The king of parabolicty"

A constructed degree-7 polynomial to have 4 (ir)rationally indifferent fix points.

- a parabolic one at -1+0i (f'=+1)
- a Siegel fix-point with a critical point in its boundary at the origin (\( f`=e^{2i\cdot \pi\cdot \alpha_{bounded}} \), value defined below)
- a (probable) Siegel fix-point with rotation number of unbounded type at 1+0i (\( f`=e^{2i\cdot \pi\cdot \alpha_{unbounded}} \))
- a Cremer fix point in the upper right quadrant at 1+i (\( f`=e^{2i\cdot \pi\cdot \alpha_{cremer}} \))

following the description in

https://fractalforums.org/fractal-mathematics-and-new-theories/28/julia-sets-true-shape-and-escape-time/2725/msg22939#msg22939. In short, a degree-7 polynomial with 8 coefficients was used to find a solution of the system consisting of 4 equations for the desired fixpoints and 4 for the derivates using maxima.

The approximative polynomial used:

\[

f(z)=(0.08742572471696199+0.9961710408648288 i)*z+

(-0.4544465364046477-2.345728439042568 i)*z^2 +

(2.849251631498275-2.082596937337672 i)*z^3 + \\

(1.259747964272526+4.46270310026291 1)*z^4 +

(-2.609925545684205+0.9479269742586326 i)*z^5 +

(-0.8053014278678783-2.116974661220343 i)*z^6 + \\

(0.6732481894689685+0.1384989222142105 i)*z^7

\]

The image below is a rational impression, point-sampled (25000 max it) in the 4-square, cropped. non-rounding controlled with approximations to the derivatives (and hence neither truely a Cremer nor Siegel disc. The fix points are surrounded by the crossing of two vertical and two horizontal lines.

The picture displays the cusp-like shape of the parabolic basin, the two Siegel discs that look rather similar to me (bound/unbounded) and the Cremer-like impression with a tongue-like feature, which I suspect vanishes with higher iteration count, so might actually be a rational value for the derivative that lies outside the 8D parameter space for this type of combination of what the critical points do.

**Values**\( \alpha_{bounded} \): The value of the continued fraction 4+1/(4+1/(4+1/(... which is of bounded type and hence has a Siegel disc with a critical point in its boundary for any polynomial with that rotation number (Zhang-theorem in A Cheritat, P Roesch,

*Herman's condition and Siegel discs of bicritical polynomials*). Here the first 100 fractions were computed, resulting in ~4.2360679774997898050514777.

\( \alpha_{unbounded} \): The value of the continued fraction 1+1/(3+1/(4+1/(5+... I do not know whether this results in a Siegel disc or a Cremer point (I suspect the former, as the denominator are growing slowly), but it is a different arithmetic condition on the rotation number. Calculated were again the first 100 fractions, resulting in ~1.3160892412682212437857742.

\( \alpha_{cremer} \): I used the value from post #39 above for the first 5 fractions, resulting in ~2.444468546637744044147-

Two consecutive continued fractions (lying on different sides of the true value) are closer than 10^-70 (arbitrarily chosen value).

The symbolic solution to the coefficients is (where m2,m3,m4 are the derivatives for the irrational fix points (bounded, unbounded, Cremer))::

`A7 = -((-8*m4)+i*((-6*m4)+50*m2-44)+25*m3-17)/100`

A6 = ((-2*m4)+i*((-14*m4)+50*m3+100*m2-136)+25*m3-100*m2+77)/100

A5 = ((-16*m4)+i*((-12*m4)+100*m2-88)+75*m3+100*m2-159)/100

A4 = -((-4*m4)+i*((-28*m4)+100*m3+200*m2-272)+25*m3-200*m2+179)/100

A3 = -((-4*m4)+i*((-3*m4)+25*m2-22)+25*m3+100*m2-121)/50

A2 = ((-m4)+i*((-7*m4)+25*m3+50*m2-68)-50*m2+51)/50

A1 = m2

A0 = 0