Uniformization of Julia sets and lacunary series

Question

If the complement of a Julia set of quadratic polynomial z^2+c is locally connected and simply connected, it is uniformized by the complement of the unit disk. Consider the uniformization map and its expansion into series. On general grounds, this map must have the unit disk as its natural boundary, i.e. it cannot be continued past unit circle. Therefore, the series expansion must be lacunary.

What are the known facts about the asymptotic behaviour of this function near its natural boundary? I am especially interested in connecting the combinatorics of the series and the geometry of the Julia set ( and the combinatorics of the number c, such as CF expansions of its real/imaginary parts, and perhaps renormalization of the poylnomial z^2+c, and also the combinatorics of multiscale analysis of the Green's function of the Schrodinger operator associated to the Julia set).

There is abundant literature on lacunary series ( see e.g. refs in Kahane "A CENTURY OF INTERPLAY BETWEEN TAYLOR SERIES, FOURIER SERIES AND BROWNIAN MOTION"), but it does not mention this example. The only mention of the lacunarity of the uniformizing map that I know about is in QUADRATIC SIEGEL DISKS WITH SMOOTH BOUNDARIES by Buff and Cheritat.

EDIT: as it is pointed out in the answer by Alexandre Eremenko, form the fact that there is natural boundary it does not follow "on general grounds" that the series is lacunary. So it should be "series" instead of "lacunary series" everywhere in the text.

Answer 1

The main premises of your question are wrong: 1. The complement of the Julia set is open, and every open set is evidently locally connected. 2. It is true that for most cases, the uniformizing function of the basin of infinity has the unit circle as a natural boundary. But this does not imply that it is a lacunary series. (Only the converse is true: lacunary series, under some assumptions about the lacunes have the circle of convergence as a natural boundary, but not vise versa!). In most cases the uniformizing function of the domain of attraction of infinity is not lacunary.

When the Julia set is connected, the uniformizing function satisfies the Bottcher equation which gives you a recurrent formula for the coefficients.

The coefficients of the uniformizing function of the Mandelbrot set have an almost explicit expression and they have been studied, for example, in

MR0937034 Levin, G. M. Arithmetic properties of a sequence of polynomials. (Russian) Uspekhi Mat. Nauk 43 (1988), no. 1(259), 203–204; translation in Russian Math. Surveys 43 (1988), no. 1, 245–246

and I suppose the same methods can be used for the uniformizing function of the basin of infinity.