Please use this identifier to cite or link to this item: http://hdl.handle.net/10016/6552

 Google™ Scholar. Others By: Dodson, M. M. - Melián, M. Victoria - Pestana, Domingo - Velani, S.L.
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 Title: Patterson measure and ubiquity Author(s): Dodson, M. M.Melián, M. VictoriaPestana, DomingoVelani, S.L. Publisher: Academia Scientiarum Fennica Issued date: 1995 Citation: Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, 1995, vol. 20, p. 37-60 URI: http://hdl.handle.net/10016/6552 ISSN: 1239-629X (Print)1798-2383 (Online) Description: 24 pages, no figures.-- MSC1991 codes: Primary 11K55; Secondary 11K60, 11F99.MR#: MR1304105 (95j:11074)Zbl#: Zbl 0816.11043 Abstract: Let $L$ be a closed subset of $\bbfR\sp k$, with Hausdorff dimension $\delta$, which supports a probability measure $m$ for which the $m$- measure of a ball of radius $r$ and centred at a point in $L$ is comparable to $r\sp \delta$. By extending the notion of ubiquity from $k$-dimensional Lebesgue measure to $m$, a natural lower bound for the Hausdorff dimension of a fairly general class of $\limsup$ subsets of $L$ is obtained. This is applied to Patterson measure supported on the limit set of a convex co-compact group to obtain the Hausdorff dimension of the set of 'well-approximable' points associated with the limit set of a convex co-compact group. The equivalent geometric result in terms of geodesic excursions on the quotient manifold is also obtained. These results are counterparts of the Jarník's theorem on simultaneous diophantine approximation. Sponsor: Research partially supported by the Royal Society European Programme. Review: PeerReviewed Publisher version: http://www.acadsci.fi/mathematica/Vol20/dodson.html Keywords: Geometrically finite groupsHausdorff dimensionUbiquityPatterson measureGeodesic excursionsJarník's theoremSimultaneous diophantine approximation Rights: © Academia Scientiarum Fennica Appears in Collections: DM - GAMA - Artículos de Revistas