Newton dynamics for entire functions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a nonconstant entire function $f$, Newton iteration uses the meromorphic map \[N_f(z)=z-\frac{f(z)}{f'(z)}.\] At poles the map is considered in the Riemann sphere; its dynamics concerns the forward iterates where these are defined. A transcendental entire function is an entire function that is not a polynomial.

Statement recorded in the supplied source.

Describe the dynamics of Newton iteration for general classes of transcendental entire functions. The target is a general dynamical description, allowing the choice and analysis of natural classes of functions. [S2]

Short English statement

Develop a dynamical description of Newton’s method for general classes of transcendental entire functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300066 (AMR-052-0066). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld, Mikhail Lyubich and contributors, Problems in Holomorphic Dynamics, arXiv:math/9205209v1 (1992), Robert Devaney, "Open Questions in Non-Rational Complex Dynamics", §5, eighth Problem, printed p. 36. https://arxiv.org/abs/math/9205209v1.