On Dynamics of Chebyshev’s Method for Quartic Polynomials

• Tarakanta Nayak ^[1] ; Soumen Pal ^[1]
  1. [1] Indian Institute of Technology, Bhubaneswar
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Let p be a normalized (monic and centered) quartic polynomial with non-trivial symmetry group. It is already known that if p is unicritical, with only two distinct zeros 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., Pal, S, Mediterr. J.Math. 22(1), 12 (2025)). 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 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 Ca 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 zeros 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 zeros of pa are unbounded and those corresponding to ±i √|a| are simply connected.

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