Dynamics of exponential-trigonometric entire maps

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.

For complex parameters $\lambda$, describe the dynamics of both $f_\lambda(z)=\lambda e^z\sin z$ and $g_\lambda(z)=\lambda e^z\cos z$. [S2]

Short English statement

Describe the iteration dynamics of the exponential-times-sine and exponential-times-cosine families as their complex parameter varies.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300060 (AMR-052-0060). 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), second problem in Section 1, pp. 35; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.