Area of filled Julia sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:24:38Z http://mathoverflow.net/feeds/question/38114 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38114/area-of-filled-julia-sets Area of filled Julia sets lhf 2010-09-08T23:38:09Z 2012-07-13T14:26:23Z <p>The recent question <a href="http://mathoverflow.net/questions/37229/area-of-the-boundary-of-the-mandelbrot-set" rel="nofollow">http://mathoverflow.net/questions/37229/area-of-the-boundary-of-the-mandelbrot-set</a> prompted me to ask this question.</p> <p>There has been some work on estimates for the area of the Mandelbrot set, e.g., a <a href="http://www.springerlink.com/content/g08r87rm6660x500/" rel="nofollow">paper</a> by John H. Ewing and Glenn Schober in <em>Numerische Mathematik</em>. Is there similar work for the estimating the area of quadratic filled Julia sets as a function of the parameter $c$? Perhaps the material in the book <a href="http://www.springer.com/mathematics/numerical+and+computational+mathematics/book/978-3-540-68546-3" rel="nofollow">Computability of Julia Sets</a> implies some estimates but I don't have the book handy and from what I recall of skimming it, there was none.</p> http://mathoverflow.net/questions/38114/area-of-filled-julia-sets/80769#80769 Answer by lhf for Area of filled Julia sets lhf 2011-11-12T19:38:36Z 2012-07-13T14:26:23Z <p>This paper contains some information about the area of filled Julia sets, though not a formula:</p> <blockquote> <p>Yang, Guoxiao, <a href="http://dx.doi.org/10.1080/02781070290013811" rel="nofollow">Some geometric properties of Julia sets and filled-in Julia sets of polynomials</a>. Complex Var. Theory Appl. 47 (2002), no. 5, 383–391. <a href="http://www.ams.org/mathscinet-getitem?mr=1906990" rel="nofollow">MR1906990 (2003c:37067)</a></p> </blockquote> <p>There is also this more promising paper by the same author, but it is in Chinese and I can't get a copy anyway:</p> <blockquote> <p>Yang, Guo Xiao, The area and diameter of filled-in Julia sets and Mandelbrot sets. Acta Math. Sinica 38 (1995), no. 5, 607–613. <a href="http://www.ams.org/mathscinet-getitem?mr=1372560" rel="nofollow">MR1372560 (96m:30040)</a></p> </blockquote> <p>If someone knows these papers, I'd be grateful for any insights.</p> <p>Problem A-1 in Milnor's <em>Dynamics in one complex variable</em> contains a formula for the area expressed as a series based on Gronwall's area theorem: $$\pi (1 - |a_2|^2 - 3|a_4|^2 - 5|a_6|^2 - \cdots)$$ The series is said to converge slowly. The coefficients of the series can be easily computed recursively though by solving $$\psi(w^2) = \psi(w)^2+c$$ for $$\def\F#1{\frac{a_{#1}}{w^{#1}}} \psi(w) = w(1 + \F2 + \F4 + \F6 + \cdots)$$</p>