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

  • $\begingroup$ Maybe it will help : "Every Misiurewicz point c is the landing point of a finite non-zero number of parameter rays at preperiodic angles. These angles are exactly the external angles of the dynamic rays which land at the critical value in the dynamic plane of c." Rational Parameter Rays of the Mandelbrot Set by Dierk Schleicher $\endgroup$ – Adam Jan 20 '15 at 20:51

"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 )


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  • $\begingroup$ This paper suggests that is is "not hard" to find the other external angles. A sketch of a method might be: consider the pre-periodic angle $.001(010)$ as the limit of periodic angles $.(0), .(00), .(001), .(001 0), .(001 01), .(001 010), .(001 010 0), .(001 010 01), .(001 010 010), \ldots$ and use the "other periodic angle" algorithm. Then the other periodic angles converge to another (pair of) pre-periodic angle(s), hopefully the ones we want. $\endgroup$ – Claude Jan 20 '15 at 22:04

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