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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

Citations:

Zbl 0593.00013