Fractals/island wake

The biggest island of the p/q wake
 * for p/q <= 1/3
 * is located on the first branch ( reading from the left to right)
 * has period = q+1
 * external rays ?
 * for p/q > 1/3
 * is located on the last branch ( reading from the left to right)
 * has period = q+2

=Examples=

Wake 1/2
period 3 island is the biggest island of wake 1/2

find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7)

Plane description : -1.76733 +0.00002 i @ 0.05

One can check it using program Mandel by Wolf Jung :

The angle 3/7  or  p011 has  preperiod = 0  and  period = 3. The conjugate angle is 4/7  or  p100. The kneading sequence is AB*  and the internal address is  1-2-3. The corresponding parameter rays are landing at the root of a primitive component of period 3.

See also:
 * $$F_{1/2}$$ family: real slice of Mandelbrot set.
 * periodic part: period doubling cascade. Escape route 1/2
 * the Myrberg-Feigenbaum point of $F_{1/2}$ family
 * chaotic part main antenna is a shrub of $$F_{1/2}$$ family

1/3
Period 4 island
 * Center X  -0.15710375803
 * Center Y  +1.03258348530
 * Pixel step +0.0000375

12/25
wake 12/25
 * root c = -0.738203140939397 +0.124839088573366 i
 * The 12/25-wake of the main cardioid is bounded by the parameter rays with the angles
 * 11184809/33554431 or  p0101010101010101010101001  and
 * 11184810/33554431 or  p0101010101010101010101010.
 * the center of the satellite component c = -0.739829393511579 +0.125072144080321 i    period = 25

biggest island of 12/25 wake
 * cardioid
 * center c = -0.744245042107463 +0.127908444364520 i
 * period = 27
 * cusp

=Size=
 * miniset-and-embedded-julia-size-estimates
 * Windows of periodicity scaling by E Demidov
 * J.A.Yorke, C.Grebogi, E.Ott, and L.Tedeschini-Lalli  "Scaling Behavior of Windows in Dissipative Dynamical Systems" Phys.Rev.Lett. 54, 1095 (1985)
 * B.R.Hunt, E.Ott Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map  J.Phys.A 30 (1997), 7067.

Haskell program

=References=