Measure and dimension of transcendental 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.

An entire function is a holomorphic map $f:\mathbb C\to\mathbb C$. Its iterates are $f^0=\mathrm{id}$ and $f^{n+1}=f\circ f^n$. Its Fatou set is the open set where the family of iterates is normal (in the spherical metric), and its Julia set $J(f)$ is the complement. The exponential and trigonometric parameter families here are \[E_\lambda(z)=\lambda e^z,\quad S_\lambda(z)=\lambda\sin z,\quad C_\lambda(z)=\lambda\cos z,\qquad \lambda\in\mathbb C\setminus\{0\}.\]Devaney describes parameter hairs as the Cantor families of curves in the complex parameter plane along which the corresponding map has Julia set equal to $\mathbb C$. Measure means planar Lebesgue measure, and dimension means Hausdorff dimension. These are hairs in parameter space, distinct from Cantor bouquets in the dynamical $z$-plane.

Statement recorded in the supplied source.

Determine the measure and Hausdorff dimension of the parameter hairs for each of $E_\lambda$, $S_\lambda$ and $C_\lambda$. [S2]

Short English statement

Determine how large the parameter hairs are, in measure and Hausdorff dimension, for all three exponential, sine and cosine families.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300065 (AMR-052-0065). 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", §4, seventh Problem, printed p. 36. https://arxiv.org/abs/math/9205209v1.