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Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

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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.

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Correspondence to Cristobal Rojas.

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Communicated by Stephen Cook.

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C.R was partially supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 731143, by project FONDECYT Regular No 1190493 and project Basal PFB-03 CMM-Universidad de Chile. M.Y. was partially supported by NSERC Discovery grant.

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Rojas, C., Yampolsky, M. Real Quadratic Julia Sets Can Have Arbitrarily High Complexity. Found Comput Math 21, 59–69 (2021). https://doi.org/10.1007/s10208-020-09457-w

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  • DOI: https://doi.org/10.1007/s10208-020-09457-w

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