Symmetries of fractals (Q1914966)

Let \(P_{c, k}(z)= z^k+ c\) and \(P^n_{c, k}\) its \(n\)th iterate. Then the classical Mandelbrot set is defined by \(M(2)= \{c\in \mathbb{C}: \{P^n_{c, 2}(0)\}_{n\in \mathbb{N}}\) remains bounded\}, i.e., one looks at the iterative behaviour of the only ``free'' critical point, this means zero of \(P_{c, 2}'\). Similarly, one can define \(M(k)\). Extending a result of \textit{C. Alexander}, \textit{I. Giblin} and \textit{D. Newton} [ibid. 14, No. 2, 32-38 (1992; Zbl 0752.58020)], in the paper under review, it is shown (Theorems 1, 2) that for \(k\geq 2\) the group of G-symmetries of \(M(k)\) equals the dihedral group of order \(k- 1\), and that for \(k\geq 3\) the group of symmetries of \(M(k)\) equals the group of G-symmetries, hence the dihedral group again. (Here the rotations and reflections of \(\mathbb{C}\), which remain the origin fixed, form the group of G-symmetries and the group of symmetries consists of all (Euclidean) motions of \(\mathbb{C}\).) The proof is by showing the following (intuitively clear) property of the subset \(FL(k)\) of \(M(k)\), which is formed by those \(c\) for which \(P_{c, k}\) has (besides \(\infty\)) an attractive fixpoint: \(\partial FL(k)\) has the same set of points closest to 0 as does \(\partial M(k)\). The further paragraphs discuss in some detail symmetries of Mandelbrot and Julia sets for arbitrary polynomials.