Computing the Laurent Series of the Map Ψ: C − [formula omitted] → C − M

Abstract

We discuss several interesting properties of the Laurent series of Ψ : C − D → C − M, the inverse of the uniformizing map of the Mandelbrot set M = { c ∈ C : c, c 2 + c,( c 2 + c) 2 + c, . , ↛ ∞ as n → ∞}. Continuity of the Laurent series on ∂ D implies local connectivity of M, which is an open question. We show how the coefficients of the series can he easily computed by following Hubbard and Douady′s construction of the uniformizing map for M. As a result, we show that the coefficients are rational with powers of 2 in their denominator and that many are zero. Furthermore, if the series is continuous on ∂D, we show that it is not Hölder continuous. We also include several empirical observations made by Don Zagier on the growth of the power of 2 in the denominator.