I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $J_r(f)=\{z\in J(f):f^n(z)\not\to\infty\}$ because in some interesting cases it is the same as the "radial Julia set'' of $f$.

For two types functions, including the exponential family of $\exp(z)-2$, I have recently shown that $J_r(f)$ is not only totally disconnected, but is zero-dimensional in the topological sense: **my paper**. The Julia sets of these functions have a relatively simple "Cantor bouquet'' structure which was used in the proofs.

**Question.** Can $J_r(f)$ contain non-degenerate or unbounded connected sets?