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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
Affiliation:
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland. e-mail: zoltan.balogh@math.unibe.ch, reto.berger@math.unibe.ch
RETO BERGER
Affiliation:
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland. e-mail: zoltan.balogh@math.unibe.ch, reto.berger@math.unibe.ch
ROBERTO MONTI
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste, 63 35121 Padova, Italy. e-mail: monti@math.unipd.it
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: tyson@math.uiuc.edu

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
Copyright
Copyright © Cambridge Philosophical Society 2010

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