Julia Sets of Rational Maps with Rotational Symmetries

By the symmetry of the Julia set of a polynomial, we mean a Euclidean isometry preserving the Julia set. Each such symmetry is, in fact, a rotation about the centroid of the polynomial. This article conducts a survey of the symmetries of polynomial Julia sets. The Euclidean isometries, which preserve the Julia set of rational maps, are then considered. A rotation preserving the Julia set of a rational map is called a rotational symmetry of its Julia set. A sufficient condition is provided for a rational map to have rotational symmetries whenever the rational map has an exceptional point. Two classes of rational maps are provided whose Julia sets have rotational symmetries of finite orders. Using this, it is proved that \( z\mapsto \mu z\) $z\mapsto \mu z$ where \(\mu ^{m+n}=1\) ${\mu }^{m+n}=1$ is a rotational symmetry of the McMullen map \( z^m+\frac{\lambda }{z^n}\) $z^m+\frac{\lambda }{z^n}$ for all m, n with \(m\ge 2\) $m\ge 2$ and \(\lambda \in \mathbb {C}\setminus \{0\}\) $\lambda \in \mathbb{C}\backslash \{0\}$ . Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynomial coincide with that of its Newton’s method and Chebyshev’s method under certain assumptions.