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Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group

Published online by Cambridge University Press:  22 June 2005

Zoltán M. Balogh  and
Jeremy T. Tyson
Zoltán M. Balogh
Affiliation:
Department of Mathematics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland. E-mail: [email protected]
Jeremy T. Tyson
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: [email protected]
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Abstract

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.

Keywords

Heisenberg groupHausdorff dimensioniterated function system22E3028A78 (primary)26A1828A78 (secondary)
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 91 , Issue 1 , July 2005 , pp. 153 - 183
Copyright
2005 London Mathematical Society

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