Braverman & Yampolsky have shown that there exist noncomputable Julia sets, i.e., there exist $c \in \mathbb{C}$ such that the Julia set of $f(z) = c + z^2$ is not computable. "A set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision."

Braverman, Mark, and Michael Yampolsky. "Non-computable Julia sets."

Journal of the American Mathematical Society(2006): 551-578. (PDF download.)

My questions are:

. Is an explicit such $c$ known? A computable $c$?Q

It seems likely these questions are answered, perhaps in the cited paper. If anyone is familiar enough with this line of work to answer, I'd appreciate it.

**Answered**. The question is answered in the paper Igor identified, particularly in its full version:

Braverman, Mark, and Michael Yampolsky. "Computability of Julia sets." arXiv link. 2007.

They prove there exist computable $c \in \mathbb{C}$ such that the Julia set of $c + z^2$ is not algorithmically computable, and provide an algorithm for computing such a $c$. Under the assumption of a complex dynamics conjecture (due to Buff & Chéritat), they obtain a polynomial-time algorithm for computing such a $c$, i.e., $n$ bits of $c$ can be computed in time polynomial in $n$.

No *explicit* $c$ is known, as far as I can tell.
(Their algorithms would not be easy to implement.)