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p/q-wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of Mandelbrot set main cardioid (period 1 hyperbolic component).

Are there any parameter plane points c from the from exterior of Mandelbrot set that do not belong to any wake ?

Wakes of Mandelbrot Set to Period 10 Image from commons

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    $\begingroup$ i.sstatic.net/BsQvr.png The proportion of external angles $[0,1)$ not covered by wakes of period $q \le P$ seems to tend to $0$ as $P \to \infty$. I think the external angle of $c$ would have to be irrational (but I doubt it's even possible), I have no proof even for the rational case... $\endgroup$
    – Claude
    Jan 15, 2023 at 11:07
  • $\begingroup$ If an external angle is rational, it's either preperiodic or periodic. if it's preperiodic, its ray lands on a Misiurewicz point and they're all in wakes; if it's periodic it lands on a parabolic point which is either attached to the period 1 cardioid, or else inside a wider wake attached to the period 1 cardioid. $\endgroup$
    – Claude
    Jan 15, 2023 at 12:31
  • $\begingroup$ @Claude Is it your diagram ? How it was computed? $\endgroup$
    – Adam
    Jan 15, 2023 at 14:36
  • $\begingroup$ I used GNUPlot with data emitted by a small Haskell program that enumerates the bulbs and totals the widths of the wakes (using exact Rational for calculations, before final conversion for output); see mathr.co.uk/web/m-primary-bulb.html which is unfortunately missing references to where I found the algorithm... $\endgroup$
    – Claude
    Jan 16, 2023 at 9:07
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    $\begingroup$ Siegel disc and Cremer points probably have irrational external angles $\endgroup$
    – Adam
    Jan 16, 2023 at 10:24

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The answer with help of the expert, Wolf Jung:

Yes, those parameters belonging to external rays that land on the Cremer and Siegel parameter points in the boundary of the main cardioid do not belong to any wake.

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