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Exceptional sets for self-similar fractals in Carnot groups

Published online by Cambridge University Press:  24 March 2010

ZOLTÁN M. BALOGH ,
RETO BERGER ,
ROBERTO MONTI  and
JEREMY T. TYSON
ZOLTÁN M. BALOGH
Affiliation:
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland. e-mail: [email protected], [email protected]
RETO BERGER
Affiliation:
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland. e-mail: [email protected], [email protected]
ROBERTO MONTI
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste, 63 35121 Padova, Italy. e-mail: [email protected]
JEREMY T. TYSON
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign 1409 West Green St., Urbana, IL 61801, U.S.A. e-mail: [email protected]
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Abstract

We consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 1 , July 2010 , pp. 147 - 172
Copyright
Copyright © Cambridge Philosophical Society 2010

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