Exceptional sets for self-similar fractals in Carnot groups

Exceptional sets for self-similar fractals in Carnot groups

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 di...

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Veröffentlicht in:Mathematical proceedings of the Cambridge Philosophical Society 2010-07, Vol.149 (1), p.147-172
Hauptverfasser: BALOGH, ZOLTÁN M., BERGER, RETO, MONTI, ROBERTO, TYSON, JEREMY T.
Format: Artikel
Sprache:eng
Schlagworte:
Estimates
Euclidean space
Fractal analysis
Fractals
Mathematical analysis
Self-similarity
Similarity
Translations
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Zusammenfassung: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.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004110000083