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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.

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.

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.

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0x11111
0x11111
  • 593
  • 2
  • 6

Uniformization of Julia sets and lacunary series

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.