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Patterson measure and ubiquity

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URI: http://hdl.handle.net/10016/6552
ISSN: 1239-629X (Print)
ISSN: 1798-2383 (Online)
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patterson_pestana_aasfm_1995.pdf (163.92 KB)
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1995
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Academia Scientiarum Fennica
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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.
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24 pages, no figures.-- MSC1991 codes: Primary 11K55; Secondary 11K60, 11F99.
MR#: MR1304105 (95j:11074)
Zbl#: Zbl 0816.11043
Keywords
Geometrically finite groups, Hausdorff dimension, Ubiquity, Patterson measure, Geodesic excursions, Jarník's theorem, Simultaneous diophantine approximation
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Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, 1995, vol. 20, p. 37-60
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