## Algorithms for computing angles in the Mandelbrot set.(English)Zbl 0603.30030

Chaotic dynamics and fractals, Proc. Conf., Atlanta/Ga. 1985, Notes Rep. Math. Sci. Eng. 2, 155-168 (1986).
[For the entire collection see Zbl 0593.00013.]
For the polynomial $$f_ c=z^ 2+c$$ let $$K_ c$$ denote the filled-in Julia set and let M be the Mandelbrot set $$\{$$ c; $$K_ c$$ is connected$$\}$$. Let $$\phi_ c$$ be the Riemann map from $${\hat {\mathbb{C}}}\setminus K_ c$$ to $${\hat {\mathbb{C}}}\setminus D(0,1)$$, normalized by $$\phi (\infty)=\infty$$, $$\phi '(\infty)>0$$, and let $$\phi_ M$$ be the corresponding map from $${\hat {\mathbb{C}}}\setminus M$$ to $${\hat {\mathbb{C}}}\setminus D(0,1)$$. Then $$\phi_ M(c)=\phi_ c(c)$$ for $$c\in {\hat {\mathbb{C}}}\setminus M$$. For each way of access to z in $$\partial K_ c$$ one can define one value of Arg $$\phi$$ $${}_ c(z)$$, and this is defined to be the external argument of z with respect to c, Arc(c,z). The external ray R(c,$$\vartheta)$$ is $$\{$$ $$z: Arg(c,z)=\vartheta \}.$$
Similar definitions may be made for external arguments in the Mandelbrot set and for certain boundary points of M (the Misiurewicz points) one has $$Arg(M,c)=Arc(c,c)$$. For these points and for the ”roots of the hyperbolic components of $$\overset\circ M$$” the external arguments are rational numbers which may be computed if one understands the iteration theory of $$f_ c$$ for the corresponding value of c.
From the results sketched it follows that R(M,$$\vartheta)$$ ends on the boundary of the central cardioid of M for a set $$\vartheta$$ of measure 0. It is also possible to calculate the external arguments of the Feigenbaum point in M. There is also some discussion of the structure of $$\partial K_ c$$ at fixed points $$z_ 0$$ which are approached in a spiral manner by external rays.
Reviewer: I.N.Baker

### MSC:

 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Zbl 0593.00013