# Rounding errors in images of Julia sets

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]
• Cross-posted in mathoverflow.net/questions/200161/…. – lhf Mar 16 '15 at 17:35
• 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. – Federico Poloni Mar 16 '15 at 19:23
• Backward stability of the iteration function is the relevant concept here. – Kirill Mar 16 '15 at 19:32
• Related: [Images of Julia sets that you can trust](webdoc.sub.gwdg.de/ebook/serien/e/IMPA_A/721.pdf), De Figuereido et al. – Federico Poloni Dec 15 '17 at 17:28
• @FedericoPoloni, now published as Rigorous bounds for polynomial Julia sets in Journal of Computational Dynamics. – lhf Jan 3 '20 at 12:17