In the Mandelbrot set for the quadratic polynomial $z \to z^2 + c$, rational external angles with even denominator are pre-periodic and have corresponding external rays which land at Misiurewicz points. For example, the external ray at external angle $\frac{1}{2}$ ($.1(0)$ in binary) lands at $-2$, and the two external rays at external angles $\frac{5}{12}$ and $\frac{7}{12}$ ($.01(10)$ and $.10(01)$ in binary) both land at $-1.543689012692\ldots$. I have two questions:
Given a rational external angle with even denominator, is there a combinatorial algorithm for determining how many external rays land on the corresponding Misiurewicz point?
Further, is there a combinatorial algorithm for determining the external angles of the other rays, if they exist?
By combinatorial I mean avoiding numerical methods like tracing external rays; I'd like something similar to the algorithm I use for the odd-denominator case (with two rays landing at the root of a hyperbolic component), going from one external angle to an angled internal address and then back to a pair of external angles (see http://arxiv.org/abs/math/9411238 ).
Some more background context and some numerical experiments that I performed can be found in my blog post: http://mathr.co.uk/blog/2015-01-15_external_angles_of_misiurewicz_points.html
"Lemma 4.1 (Number of Rays at Misiurewicz Points) Suppose that a preperiodic angle θ has preperiod l and period n. Then the kneading sequence K(θ) has the same preperiod l, and its period k divides n. If n/k > 1, then the total number of parameter rays at preperiodic angles landing at the same point as the ray at angle θ is n/k; if n/k = 1, then the number of parameter rays is 1 or 2. " ( Rational Parameter Rays of the Mandelbrot Set Dierk Schleicher )
HTH