Are there any points on the parameter plane that do not belong to any wake?

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 ?

Image from commons

• 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... Jan 15, 2023 at 11:07
• 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. Jan 15, 2023 at 12:31
• @Claude Is it your diagram ? How it was computed?
– Adam
Jan 15, 2023 at 14:36
• 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... Jan 16, 2023 at 9:07
• Siegel disc and Cremer points probably have irrational external angles
– Adam
Jan 16, 2023 at 10:24

1 Answer

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.