Summary
We obtain upper bounds for the area of the Mandelbrot set. An effective procedure is given for computing the coefficients of the conformal mapping from the exterior of the unit circle onto the exterior of the Mandelbrot set. The upper bound is obtained by computing finitely many of these coefficient and applying Green's Theorem. The error in such calculations is estimated by deriving explicit formulas for infinitely many of the coefficients and comparing.
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References
Douady, A., Hubbard, J. (1982): Itération des polynômes quadratiques complexes. C. R. Acad. Sci. Paris294, 123–126
Ewing, J.H., Schober, G. (1990): On the coefficients of the mapping to the exterior of the Mandelbrot set. Mich. Math. J.37, 315–320
Gronwall, T.H. (1914–15): Some remarks on conformal representation. Ann. of Math.16, 72–76
Jungreis, I. (1985): The uniformization of the complemen of the Mandelbrot set. Duke Math. J.52, 935–938
Levin, G.M. (1988): On the arithmetic properties of a certain sequence of polynomials. Russian Math. Surveys,43, 245–246
Milnor, J. (1989): Self-similarity and hairiness in the Mandelbrot set. In: M. Tangora, ed. Computers in Geometry and Topology. Lec. Notes Pure Appl. Math.114, Dekker, New York, pp. 211–257
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Partially supported by a grant from the National Science Foundation