Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps

TL;DR

This paper studies a family of transcendental entire functions with zeros, revealing how their Newton's method can produce wandering and Baker domains, advancing understanding of complex dynamics in Newton maps.

Contribution

It introduces a systematic analysis of projectable functions and extends the logarithmic lifting method to characterize Newton maps with wandering domains.

Findings

Existence of wandering domains in the Newton maps of the family

Parameter regions with wandering and Baker domains identified

Extension of the logarithmic lifting method to finite-type maps

Abstract

We present a one-parameter family $F_\lambda$ of transcendental entire functions with zeros, whose Newton's method yields wandering domains, coexisting with the basins of the roots of $F_\lambda$. Wandering domains for Newton maps of zero-free functions have been built before by, e.g., Buff and R\"uckert based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions $f(z)$ in the complex plane that are semiconjugate, via the exponential, to some map $g(w)$, which may have at most a countable number of essential singularities. In this paper we make a systematic study of the general relation (dynamical and otherwise) between $f$ and $g$, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those $g$ of finite-type. We apply these results to characterize the entire functions with zeros whose Newton's method projects to some map $g$ which is defined at both $0$ and $\infty$. The family $F_\lambda$ is the simplest in this class, and its parameter space shows open sets of $\lambda$-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton's root-finding method fails.