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Proceedings of the London Mathematical Society 2005 91(1):153-183; doi:10.1112/S0024611504015205
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© London Mathematical Society

Hausdorff Dimensions of Self-Similar and Self-Affine Fractals in the Heisenberg Group

Zoltán M. Balogh and Jeremy T. Tyson

Department of Mathematics, University of Bern Sidlerstrasse 5, 3012 Bern, Switzerland. E-mail: zoltan.balogh{at}math-stat.unibe.ch
Department of Mathematics, University of Illinois 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: tyson{at}math.uiuc.edu

Received 16 March 2004. Revision received 9 September 2004.

We study the Hausdorff dimensions of invariant sets for self-similar and self-affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self-affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra-Cassano. 2000 Mathematics Subject Classification 22E30, 28A78 (primary), 26A18, 28A78 (secondary).

Key Words: Heisenberg group • Hausdorff dimension • iterated function system

Z. M. B. was supported by a grant from the Swiss NSF.

J. T. T. was supported by NSF grant DMS 0228807. The research for this paper was done while J. T. T. was a visitor at the University of Berne during 2003. He wishes to thank the department for its hospitality.


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