The inverse problem for Julia limiting directions in the Eremenko--Lyubich class

We study the inverse problem for Julia limiting directions in the Eremenko-Lyubich class $\mathcal B$, which is the class of transcendental entire functions with a bounded set of singular values, and its meromorphic generalization $\mathcal B_{mer}$. Every nonempty compact subset of the circle can be the set of Julia limiting directions of an entire $f\in\mathcal B$. For finite unions of nondegenerate closed arcs, the corresponding $f\in\mathcal B$, even with every finite order at or above an explicit bound determined by the arc lengths, can be attained. We also refine the series construction of the latter two authors to obtain $f\in \mathcal B_{mer}$ with any prescribed order and the set of Julia limiting directions.