Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

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.
9 pages, 1 figure. To be published in the journal Found. Comp. Math. (FoCM). arXiv admin note: text overlap with arXiv:1703.04660