How to calculate the Feigenbaum constant to high precision? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:42:59Z http://mathoverflow.net/feeds/question/47421 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47421/how-to-calculate-the-feigenbaum-constant-to-high-precision How to calculate the Feigenbaum constant to high precision? Andrej Bauer 2010-11-26T12:17:09Z 2011-04-30T15:34:45Z <p>I am looking for a reference on how to calculate the <a href="http://en.wikipedia.org/wiki/Feigenbaum_constants" rel="nofollow">Feigenbaum constant</a> $\delta$ to high precision. It seems that naive methods do not work well because they lead to solving very high degree polynomials.</p> <p>I have searched Math Reviews, Googled around, and the best I found was Appendix A of <a href="http://keithbriggs.info/documents/Keith_Briggs_PhD.pdf" rel="nofollow">Keith Brigg's thesis</a>. Wikipedia gives <a href="http://pi.lacim.uqam.ca/piDATA/feigenbaum.txt" rel="nofollow">a link</a> which contains $\delta$ to high precision, but does not explain how it was computed.</p> http://mathoverflow.net/questions/47421/how-to-calculate-the-feigenbaum-constant-to-high-precision/47423#47423 Answer by Theo Buehler for How to calculate the Feigenbaum constant to high precision? Theo Buehler 2010-11-26T12:30:28Z 2010-11-26T12:30:28Z <p>As far as I know O.E. Lanford III was the first to compute the Feigenbaum constant around 1980 using interval arithmetics. That is to say, he represented real numbers not by floating point arithmetics but rather using intervals containing the sought number. If I remember correctly he had at least fifty decimals at some point (he told me orally, so I might be completely wrong).</p> <p>If you enter Author="Lanford" and Anywhere="Feigenbaum" in MathSciNet, you should find the relevant articles and some general remarks on using computers in rigorous proofs (I don't know how to post links here).</p> http://mathoverflow.net/questions/47421/how-to-calculate-the-feigenbaum-constant-to-high-precision/63534#63534 Answer by Jeremy Kahn for How to calculate the Feigenbaum constant to high precision? Jeremy Kahn 2011-04-30T15:34:45Z 2011-04-30T15:34:45Z <p>One idea for computing the Feigenbaum constant is as follows: $\delta$ is the unique expanding eigenvalue for period-doubling renormalization at its fixed point $F$. So take any real-analytic 1-parameter family of univalent maps that is transverse to the stable manifold of renormalization (for example, the real quadratic family). Then the iterated renormalizations of this 1-parameter family will converge to the unstable manifold of renormalization. The unstable manifold is of course mapped to itself by renormalization and the derivative at the fixed point will be $\delta$. </p> <p>It should be possible to implement this numerically, by keeping track of the power series for a real-analytic family of real-analytic univalent map, and applying the renormalization by replacing $f_\lambda$ with $f_\lambda \circ f_\lambda$ (and suitably rescaling). Because renormalization acts as a contraction on these 1-parameter families, this procedure should be computationally stable, and provide a number of digits proportional to the number of times one renormalizes. </p> <p>One classical reference for the stable/unstable manifold picture for period doubling renormalization is Iterated Maps on the Interval as Dynamical Systems by Collet and Eckmann. </p>