Smoothness of exponential-family parameter hairs

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 $J(f)$. Write $E_\lambda(z)=\lambda e^z$, $\lambda\in\mathbb C\setminus\{0\}$. The hairs in this question are the parameter-plane curves on which $J(E_\lambda)=\mathbb C$, described in the source; regularity concerns real curves, with $C^\infty$ meaning smooth of every order and analytic meaning real analytic.

Statement recorded in the supplied source.

Are these exponential-family parameter hairs $C^\infty$? Are they analytic? Where and how do they terminate? [S2]

Short English statement

Are the parameter hairs smooth to every order or analytic, and where and in what way do they end?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300062 (AMR-052-0062). 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 3 (parameter hairs), pp. 36; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.