Full-plane Julia sets in the exponential family

Definitions and mathematical statement

For a nonzero complex parameter $\lambda$, consider the transcendental entire map $E_\lambda(z)=\lambda e^z$ and its iterates. Its Fatou set $F(E_\lambda)$ is the largest open subset of $\mathbb C$ on which $\{E_\lambda^n:n\ge0\}$ is a normal family for locally uniform convergence in the spherical metric. Its Julia set is $J(E_\lambda)=\mathbb C\setminus F(E_\lambda)$.

Determine completely the parameter locus \[ \mathcal J=\{\lambda\in\mathbb C\setminus\{0\}: J(E_\lambda)=\mathbb C\}. \] The desired characterization should describe membership using parameter data, the orbit of the singular value, or another explicit dynamical criterion. Restating membership as absence of a Fatou set does not provide the requested parameter-space description.

There are no finite critical points, because $E_\lambda'(z)\ne0$; the finite asymptotic value zero is the singular value governing the family. Its forward orbit begins $0,\lambda,\lambda e^\lambda, \ldots$. Conditions on this orbit often describe important subsets of $\mathcal J$, but a full answer must also address parameters with bounded, recurrent, or nonconvergent singular orbits, rather than only those lying on known escaping curves.

The equality sought is equality as sets in the complex plane. Full Hausdorff dimension, positive or full planar measure, or a dense Julia set is not substituted for the precise condition. The excluded parameter $\lambda=0$ gives a constant map and is outside the transcendental family under discussion.

Short English statement

Characterize exactly the nonzero exponential parameters for which every complex point lies in the Julia set.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5300061 (AMR-052-0061), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stony Brook - Open Problems in Dynamical Systems, 1992, Devaney P3. Original source indexed by AMR-052-0061. https://www.math.stonybrook.edu/open-problems-dynamical-systems.
• S3 Devaney, Open Questions in Non-Rational Complex Dynamics, exponential-family question, in Bielefeld–Lyubich (1992). https://www.math.stonybrook.edu/preprints/ims92-7.pdf.