Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

Abstract

We show that there exist real parameters $$c\\in (-2,0)$$ c ∈ ( - 2 , 0 ) for which the Julia set $$J_{c}$$ J c of the quadratic map $$z^{2} + c$$ 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 $$J_{c}$$ 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.