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The recent question Area of the boundary of the Mandelbrot set ? prompted me to ask this question.

There has been some work on estimates for the area of the Mandelbrot set, e.g., a paper by John H. Ewing and Glenn Schober in Numerische Mathematik. 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 Computability of Julia Sets implies some estimates but I don't have the book handy and from what I recall of skimming it, there was none.

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  • $\begingroup$ See my answer to that question - it points to work of Buff and Cheritat for the 'latest' on that question. There is also lots of earlier work on estimating the Hausdorff dimension (when it is <2 ). $\endgroup$
    – Jacques Carette
    Commented Sep 9, 2010 at 0:39
  • $\begingroup$ I meant the area of the filled Julia sets $K$, not the Julia sets $J$. $\endgroup$
    – lhf
    Commented Sep 9, 2010 at 2:25

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This paper contains some information about the area of filled Julia sets, though not a formula:

Yang, Guoxiao, Some geometric properties of Julia sets and filled-in Julia sets of polynomials. Complex Var. Theory Appl. 47 (2002), no. 5, 383–391. MR1906990 (2003c:37067), Zbl 1028.30019

There is also this more promising paper by the same author, but it is in Chinese and I can't get a copy anyway:

Yang, Guo Xiao, The area and diameter of filled-in Julia sets and Mandelbrot sets. Acta Math. Sinica 38 (1995), no. 5, 607–613. MR1372560 (96m:30040), Zbl 0895.30018

If someone knows these papers, I'd be grateful for any insights.

Problem A-1 in Milnor's Dynamics in one complex variable 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 $$ \def\F#1{\frac{a_{#1}}{w^{#1}}} \psi(w) = w(1 + \F2 + \F4 + \F6 + \cdots) $$

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  • $\begingroup$ what mean the values $a_k$ and $\psi(w^2)=\psi(w)^2+c$? $\endgroup$
    – user128924
    Commented Sep 16, 2018 at 10:19
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    $\begingroup$ The functional equation $\psi(w^2)=\psi(w)^2+c$ has a formal power series solution in powers of $1/w$. This series is called the Bottcher coordinate. The $a_k$ are the coefficients of this power series. It converges in a neighborhood of $\infty$ and is a semi-conjugation from $x^2+c$ to the function $x^2$. $\endgroup$
    – Joe Silverman
    Commented Sep 16, 2018 at 12:10

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