The Julia sets of Chebyshev's method with small degrees
Tarakanta Nayak, Soumen Pal

TL;DR
This paper investigates the dynamics of Chebyshev's method applied to cubic polynomials, revealing conditions under which the Julia set is connected and classifying the degrees of the method.
Contribution
It determines the degree of Chebyshev's method for cubic polynomials and characterizes the connectivity of Julia sets based on polynomial and fixed point properties.
Findings
Degree of Chebyshev's method for cubic polynomials is 4, 6, or 7.
Julia set is connected for unicritical or non-generic cubics.
Connected Julia sets occur when the multiplier of an extraneous fixed point is in [-1,1].
Abstract
Given a polynomial , the degree of its Chebyshev's method is determined. If is cubic then the degree of is found to be or and we investigate the dynamics of in these cases. If a cubic polynomial is unicritical or non-generic then, it is proved that the Julia set of is connected. The family of all rational maps arising as the Chebyshev's method applied to a cubic polynomial which is non-unicritical and generic is parametrized by the multiplier of one of its extraneous fixed points. Denoting a member of this family with an extraneous fixed point with multiplier by , we have shown that the Julia set of is connected whenever .
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Taxonomy
TopicsIterative Methods for Nonlinear Equations · Mathematical and Theoretical Analysis · Mathematical Dynamics and Fractals
