Chebyshev's method for exponential maps

TL;DR

This paper characterizes when Chebyshev's method applied to entire functions results in rational maps, explores their fixed points and Julia sets, and relates their dynamics to polynomial maps, especially for functions of the form pe^{q}.

Contribution

It proves that Chebyshev's method yields rational maps only for functions of the form p(z)e^{q(z)} and analyzes their fixed points, Julia sets, and conjugacy to polynomial maps.

Findings

Chebyshev's method produces rational maps only for functions p(z)e^{q(z)}.

The Julia set of these maps is invariant under certain rotations.

For n ≤ 16, the Julia set is connected.

Abstract

It is proved that the Chebyshev's method applied to an entire function $f$ is a rational map if and only if $f(z) = p(z) e^{q(z)}$, for some polynomials $p$ and $q$. These are referred to as rational Chebyshev maps, and their fixed points are discussed in this article. It is seen that $\infty$ is a parabolic fixed point with multiplicity one bigger than the degree of $q$. Considering $q(z)=p(z)^n+c$, where $p$ is a linear polynomial, $n \in \mathbb{N}$ and $c$ is a non-zero constant, we show that the Chebyshev's method applied to $pe^q$ is affine conjugate to that applied to $z e^{z^n}$. We denote this by $C_n$. All the finite extraneous fixed points of $C_n$ are shown to be repelling. The Julia set $\mathcal{J}(C_n)$ of $C_n$ is found to be preserved under rotations of order $n$ about the origin. For each $n$, the immediate basin of $0$ is proved to be simply connected. For all $n \leq 16$, we prove that $\mathcal{J}(C_n)$ is connected. The Newton's method applied to $ze^{z^n}$ is found to be conjugate to a polynomial, and its dynamics is also completely determined.