The Julia sets of Chebyshev's method with small degrees

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 $p$, the degree of its Chebyshev's method $C_p$ is determined. If $p$ is cubic then the degree of $C_p$ is found to be $4,6$ or $7$ and we investigate the dynamics of $C_p$ in these cases. If a cubic polynomial $p$ is unicritical or non-generic then, it is proved that the Julia set of $C_p$ 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 $\lambda $ by $C_\lambda$, we have shown that the Julia set of $C_\lambda$ is connected whenever $\lambda \in [-1,1]$.