Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

Abstract

We show that there exist real parameters $$c\in (-2,0)$$ 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.