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One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function.

How do rounding errors affect the results?

I'm looking for references on this issue, especially but not exclusively for the escape time method.

The only ones I have found so far are:

  • "The [inverse iteration] method is very insensitive to round-off errors, since $f$ tends to be expanding on its Julia set, so that $f^{-1}$ tends to be contracting.'' [Milnor, Dynamics in one complex variable, page 49]
  • "Theorem 1 makes it plausible that in almost all cases rounding errors do not affect the computer graphics of Julia sets.'' [Steinmetz, Rational iteration, page 175]
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    $\begingroup$ Cross-posted in mathoverflow.net/questions/200161/…. $\endgroup$ – lhf Mar 16 '15 at 17:35
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    $\begingroup$ When one wants certified/rigorous results out of a computer, the main technique in dynamical systems is interval arithmetic. There is active research on this topic; for instance, it was only around 2000 that it was proved that the Lorenz attractor really exists and is not only a numerical artifact. $\endgroup$ – Federico Poloni Mar 16 '15 at 19:23
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    $\begingroup$ Backward stability of the iteration function is the relevant concept here. $\endgroup$ – Kirill Mar 16 '15 at 19:32
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    $\begingroup$ Related: [Images of Julia sets that you can trust](webdoc.sub.gwdg.de/ebook/serien/e/IMPA_A/721.pdf), De Figuereido et al. $\endgroup$ – Federico Poloni Dec 15 '17 at 17:28
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    $\begingroup$ @FedericoPoloni, now published as Rigorous bounds for polynomial Julia sets in Journal of Computational Dynamics. $\endgroup$ – lhf Jan 3 at 12:17
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Can it be proved that the very existance of these convergence / divergence boundaries is stable in the face of numerical rounding artifacts? They seem to bear an uncanny resemblance to quantization error and sinc functions.

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