  We show that there exist real parameters $c$ for which the Julia set $J_c$ of
the quadratic map $z^2+c$ has arbitrarily high computational complexity. More
precisely, we show that for any given complexity threshold $T(n)$, there exist
a real parameter $c$ such that the computational complexity of computing $J_c$
with $n$ bits of precision is higher than $T(n)$. This is the first known class
of real parameters with a non poly-time computable Julia set.

Dynamical Systems;Computational Complexity