The Julia sets of Chebyshev's method with small degrees

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 λ by C λ , we have shown that the Julia set of C λ is connected whenever λ ∈ [−1, 1].