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Fractal dimensions for Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach. (English) Zbl 0851.30027
Let \(\Gamma\) be a geometrically finite Kleinian group acting on the \((N + 1) \)-dimensional hyperbolic space. We suppose that \(\Gamma\) has at least one cusp which we denote by \(p\). Let \(L (\Gamma)\) be the limit set of \(\Gamma\) and let \(\delta (\Gamma)\) be its Hausdorff dimension. In this framework one can at least formulate analogues of the classical theorems of ‘metric number theory’. In this paper the author studies the sets of badly approximable limit points, that is, sets analogous to the sets of real numbers \(x\) for which \(|x - p/q |> C/q^\alpha\) for some constant \(C > 0\) and all rational numbers \(p/q\). Here \(\alpha > 2\). In this paper the author proves an analogue of the classical theorem of Jarník which determines the Hausdorff dimension of these sets but now in the context indicated above if the rank of each parabolic subgroup is less than \(\delta (\Gamma)\). The method involves a study of the details of the canonical measure on the limit set (due to Stratmann and Velani) and a fairly classical approach to the determination of the Hausdorff dimension based on a lemma of Billingsley.

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
11J83 Metric theory
Full Text: DOI
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