Symmetry and dynamics of Chebyshev's method

Abstract: The set of all holomorphic Euclidean isometries preserving the Julia set of a rational map $R$ is denoted by $\Sigma R$. It is shown in this article that if a root-finding method $F$ satisfies the Scaling theorem, i.e., for a polynomial $p$, $F_p$ is affine conjugate to $F_{\lambda p \circ T}$ for every nonzero complex number $\lambda $ and every affine map $T$, then for a centered polynomial $p$ of order at least two (which is not a monomial), $\Sigma p\subseteq \Sigma F_p$. As the Chebyshev's method satisfies the Scaling theorem, we have $\Sigma p \subseteq \Sigma {C_p}$, where $p$ is a centered polynomial. The rest part of this article is devoted to explore the situations where the equality holds and in the process, the dynamics of $C_p$ is found. We show that the Julia set $\mathcal{J}(C_p)$ of $ C_p$ can never be a line. If a centered polynomial $p$ is (a) unicritical, (b) having exactly two roots with the same multiplicity, (c) cubic and $\Sigma p$ is non-trivial or (d) quartic, $0$ is a root of $p$ and $\Sigma p $ is non-trivial then it is proved that $\Sigma p = \Sigma C_p$. It is found in all these cases that the Fatou set $\mathcal{F}(C_p)$ is the union of all the attracting basins of $C_p$ corresponding to the roots of $p$ and $\mathcal{J}(C_p)$ is connected. It is observed that $\mathcal{J}(C_p)$ is locally connected in all these cases.