Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 711-727

FROM APOLLONIAN PACKINGS TO HOMOGENEOUS SETS

Sergei Merenkov and Maria Sabitova

University of Illinois at Urbana-Champaign, Department of Mathematics
1409 W. Green Str., Urbana, IL 61801, U.S.A.; merenkov 'at' illinois.edu
and The City College of New York, Department of Mathematics
Convent Ave. at 138th Str., New York, NY 10031, U.S.A.; smerenkov 'at' ccny.cuny.edu

CUNY Queens College, Department of Mathematics
65-30 Kissena Blvd., Flushing, NY 11367, U.S.A.; Maria.Sabitova 'at' qc.cuny.edu

Abstract. We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of Boyd's theorem (relating the curvature distribution function of an Apollonian packing to its exponent and the Hausdorff dimension of the residual set) for Sierpinski carpets that are Julia sets of hyperbolic rational maps.

2010 Mathematics Subject Classification: Primary 52C26, 37F35.

Key words: Apollonian packing, Sierpinski carpet, curvature distribution function, Hausdorff dimension, hyperbolic rational map, Julia set.

Reference to this article: S. Merenkov and M. Sabitova: From Apollonian packings to homogeneous sets. Ann. Acad. Sci. Fenn. Math. 40 (2015), 711-727.

Full document as PDF file

doi:10.5186/aasfm.2015.4046