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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.
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the imp
We define the epsilon-distortion complexity of a set as the shortest program, running on a universal Turing machine, which produces this set at the precision epsilon in the sense of Hausdorff distance. Then, we estimate the epsilon-distortion complex
We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.
We give an introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions.