Real Quadratic Julia Sets Can Have Arbitrarily High Complexity

Cristobal Rojas, Michael Yampolsky

Research output: Contribution to journalArticlepeer-review


We show that there exist real parameters c∈ (- 2 , 0) for which the Julia set Jc of the quadratic map z2+ 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 Jc 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.

Original languageEnglish
JournalFoundations of Computational Mathematics
Publication statusAccepted/In press - 1 Jan 2020


  • Complexity lower bounds
  • Julia sets
  • Renormalization
  • Unimodal maps

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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