Chebyshev polynomials on generalized Julia sets

Abstract

Let $(f_n{)}_{n=1}^{\infty }$ be a sequence of nonlinear polynomials satisfying some mild conditions. Furthermore, let $F_m(z)=(f_m\circ f_{m-1}\dots \circ f_1)(z)$ and ${\rho }_m$ be the leading coefficient for $F_m$. It is shown that on the Julia set $J_{(f_n)}$, the Chebyshev polynomial of the degree deg$F_m$ is of the form $F_m(z)/{\rho }_m-{\tau }_m$ for all $m\in \mathbb{N}$ where ${\tau }_m\in \mathbb{C}$. This generalizes the result obtained for autonomous Julia sets.

Cited by in corpus (2)

• Orthogonal polynomials for the weakly equilibrium Cantor sets
• Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings