On dynamics of the Chebyshev's method for quartic polynomials

TL;DR

This paper studies the dynamics of Chebyshev's method applied to quartic polynomials, revealing properties of fixed points, basins, and the structure of Julia and Fatou sets for specific polynomial families.

Contribution

It characterizes the dynamical behavior of Chebyshev's method for a family of quartic polynomials, including fixed point nature and basin connectivity, expanding understanding beyond symmetric cases.

Findings

All extraneous fixed points are repelling.

No invariant Siegel disks or Herman rings are present.

For positive a, at least two basins are unbounded and simply connected.

Abstract

Let $p$ be a normalized (monic and centered) quartic polynomial with non-trivial symmetry groups. It is already known that if $p$ is unicritical, with only two distinct roots with the same multiplicity or having a root at the origin then the Julia set of its Chebyshev's method $C_p$ is connected and symmetry groups of $p$ and $C_p$ coincide~[Nayak, T., and Pal, S., Symmetry and dynamics of Chebyshev's method, \cite{Sym-and-dyn}]. Every other quartic polynomial is shown to be of the form $p_a (z)=(z^2 -1)(z^2-a)$ where $a \in \mathbb{C}\setminus \{-1,0,1\}$. Some dynamical aspects of the Chebyshev's method $C_a$ of $p_a$ are investigated in this article for all real $a$. It is proved that all the extraneous fixed points of $C _a$ are repelling which gives that there is no invariant Siegel disk for $C_a$. It is also shown that there is no Herman ring in the Fatou set of $C_a$. For positive $a$, it is proved that at least two immediate basins of $C_a$ corresponding to the roots of $p_a$ are unbounded and simply connected. For negative $a$, it is however proved that all the four immediate basins of $C_a$ corresponding to the roots of $p_a$ are unbounded and those corresponding to $\pm i\sqrt{|a|}$ are simply connected.