Chebyshev's method applied to polynomials with two distinct roots

TL;DR

This paper investigates the dynamics of Chebyshev's method when applied to quadratic polynomials, demonstrating connected Julia sets and describing the structure of Fatou and Julia sets in this context.

Contribution

It provides a rigorous analysis of the Julia and Fatou sets for Chebyshev's method on polynomials with two roots, including cases with equal multiplicities.

Findings

Julia set is connected for polynomials with two roots.

Fatou set comprises basins of attraction for the roots.

Boundary of basins is connected when roots have equal multiplicity.

Abstract

The Julia set of the Chebyshev's method applied to polynomials with exactly two distinct roots is shown to be connected, and its Fatou set is proved to be the union of attracting basins corresponding to the two roots. Further, if the two roots have the same multiplicity then the common boundary of the two immediate basins is proved to be a connected subset of the Julia set.