Representative transcendental entire dynamics

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.

A transcendental entire map is a holomorphic function $f:\mathbb C\to\mathbb C$ that is not a polynomial. Its dynamics means the iterates $f^n$, their orbit behaviour, and the stable (Fatou) and Julia sets: the Fatou set is the largest open set where the iterates form a normal family, and its complement is $J(f)$.

Statement recorded in the supplied source.

Find a collection of representative transcendental entire maps whose dynamics can be understood and can model general phenomena. [S2]

Short English statement

Develop representative examples that make the dynamics of transcendental entire functions understandable.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300059 (AMR-052-0059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Robert Devaney, "Open Questions in Non-Rational Complex Dynamics", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), first problem in Section 1, pp. 35; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.