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2024-11-25

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cc by (c) Robert Florido et al., 2024
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/222644

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

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https://doi.org/DOI:10.1017/prm.2024.81

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 Rückert [23] 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.

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Sistemes dinàmics complexos, Funcions meromorfes

Subject (English)

Complex dynamical systems, Meromorphic functions

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Articles publicats en revistes (Matemàtiques i Informàtica)

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FLORIDO LLINÀS, Robert and FAGELLA RABIONET, Núria. Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2024. Vol. 2024. ISSN 0308-2105. [consulted: 16 of August of 2026]. Available at: https://hdl.handle.net/2445/222644

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