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Julia sets of rational maps with rotational symmetries

Tarakanta Nayak, Soumen Pal

Abstract

By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a survey of the symmetries of polynomial Julia sets is made. Then the Euclidean isometries preserving the Julia set of rational maps are 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 μz$ where $μ^{m+n}=1$ is a rotational symmetry of the McMullen map $ z^m+\fracλ{z^n}$ for all $m,n$ with $m\geq 2$ and $λ\in \mathbb{C}\setminus \{0\}$. Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynmial coincide with that of its Newton's method and Chebyshev's method.

Julia sets of rational maps with rotational symmetries

Abstract

By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a survey of the symmetries of polynomial Julia sets is made. Then the Euclidean isometries preserving the Julia set of rational maps are 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 where is a rotational symmetry of the McMullen map for all with and . Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynmial coincide with that of its Newton's method and Chebyshev's method.
Paper Structure (7 sections, 30 theorems, 38 equations, 8 figures)

This paper contains 7 sections, 30 theorems, 38 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Symmetries of polynomial Julia sets
  3. Polynomials with the same Julia set
  4. Symmetries of rational Julia sets
  5. Rational maps with an exceptional point
  6. Two new classes of rational maps
  7. Newton's and Chebyshev's methods

Key Result

Lemma 2.1

If $R_1$ and $R_2$ are two rational maps such that $R_1=\psi^{-1}\circ R_2 \circ \psi$ for some Möbius map $\psi$ then $\mathcal{J}(R_2)=\psi(\mathcal{J}(R_1))$.

Figures (8)

  • Figure 1: Symmetries of the Julia sets
  • Figure 2: The Julia set of $z^2-1$ and $-z^2+1$
  • Figure 3: The conjugacy
  • Figure 4: Julia sets of $\frac{3z^3}{3-z^3}$ and $\frac{z^3}{1-2z^2}$
  • Figure 5: Julia set of rational maps of the form $a\frac{P(z)}{Q(z)}$
  • ...and 3 more figures

Theorems & Definitions (51)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: Beardon_book
  • Theorem 2.6
  • Remark 2.1
  • Example 2.1
  • Theorem 3.1
  • ...and 41 more