2023/09/14 by Tarakanta Nayak, Nayak, Tarakanta, Soumen Pal +1
Mathematics · #37F10 #65H05 #Dynamical Systems (math.DS) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2309.07562
openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Cp is connected and symmetry groups of p and Cp coincide~[Nayak, T., and Pal, S., Symmetry and dynamics of Chebyshev's method, \citeSym-and-dyn]. Every other quartic polynomial is shown to be of the form pa (z)=(z2 -1)(z2-a) where a ∈ ℂ∖ \-1,0,1\. Some dynamical aspects of the Chebyshev's method Ca of pa 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 Ca. It is also shown that there is no Herman ring in the Fatou set of Ca. For positive a, it is proved that at least two immediate basins of Ca corresponding to the roots of pa are unbounded and simply connected. For negative a, it is however proved that all the four immediate basins of Ca corresponding to the roots of pa are unbounded and those corresponding to ± i√(|a|) are simply connected.