Chebyshev polynomials on generalized Julia sets

TL;DR

This paper generalizes the form of Chebyshev polynomials on Julia sets to sequences of nonlinear polynomials, showing a specific structure involving iterated compositions and leading coefficients.

Contribution

It extends known results from autonomous Julia sets to non-autonomous sequences of polynomials, providing a new explicit form for Chebyshev polynomials on these sets.

Findings

Chebyshev polynomials on the Julia set have a specific form involving iterated compositions.

The result generalizes previous autonomous Julia set results.

The structure involves the leading coefficient and a complex translation term.

Abstract

Let $(f_n)_{n=1}^\infty$ be a sequence of nonlinear polynomials satisfying some mild conditions. Furthermore, let $F_m(z)=(f_m\circ f_{m-1}\ldots \circ f_1)(z)$ 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.