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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.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 11J83 Metric theory
##### Keywords:
limit set; Jarník theorem; Kleinian group; Hausdorff dimension
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##### References:
 [1] Beardon, A. F., On the Hausdorff dimension of general Cantor sets,Math. Proc. Cambridge Philos. Soc. 61 (1965), 679–694. · Zbl 0145.05502 [2] Besicovitch, A. S., Sets of fractional dimension (IV): On rational approximation to real numbers,J. London Math. Soc. 9 (1934), 126–131. · Zbl 0009.05301 [3] Billingsley, P.,Ergodic Theory and Information, Wiley & Sons, New York, 1965. · Zbl 0141.16702 [4] Denker, M. andUrbański, M., Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point,J. London Math. Soc. 43 (1991), 107–118. · Zbl 0734.28007 [5] Denker, M. andUrbański, M., Geometric measures for parabolic rational maps,Ergodic Theory Dynamical Systems 12 (1992), 53–66. · Zbl 0737.58030 [6] Denker, M. andUrbański, M., The capacity of parabolic Julia sets,Math. Z. 211 (1992), 73–86. · Zbl 0763.30009 [7] Falconer, K.,Fractal Geometry, Wiley & Sons, New York, 1990. [8] Jarník, V., Diophantische Approximationen und Hausdorff Mass,Mat. Sb. 36 (1929), 371–382. · JFM 55.0719.01 [9] Melián, M. V. andPestana, D., Geodesic excursions into cusps in finite-volume hyperbolic manifolds,Michigan Math. J. 40 (1993), 77–93. · Zbl 0793.53052 [10] Nicholls, P. J.,The Ergodic Theory of Discrete Groups, London Math. Soc. Lecture Note Ser.143, Cambridge Univ. Press, Cambridge, 1989. · Zbl 0674.58001 [11] Patterson, S. J., The limit set of a Fuchsian group,Acta Math. 136 (1976), 241–273. · Zbl 0336.30005 [12] Patterson, S. J., Lectures on measures on limit sets of Kleinian groups, inAnalytical and Geometric Aspects of Hyperbolic Space (Epstein, D. B. A., ed.), London Math. Soc. Lecture Note Ser.111, pp. 281–323, Cambridge Univ. Press, Cambridge, 1987. [13] Stratmann, B., Diophantine approximation in Kleinian groups,Math. Proc. Cambridge Philos. Soc. 116 (1994), 57–78. · Zbl 0809.30036 [14] Stratmann, B., The Hausdorff dimension of bounded geodesics on geometrically finite manifolds,Preprint in Math. Gottingensis 39 (1993), submitted toErgodic Theory Dynamical Systems. [15] Stratmann, B. andUrbański, M., The box-counting dimension for geometrically finite Kleinian groups,Preprint in Math. Gottingensis 35 (1993), to appear inFund. Math. [16] Stratmann, B. andUrbański, M., In preparation. [17] Stratmann, B. andVelani, S., The Patterson measure for geometrically finite groups with parabolic elements, new and old,Proc. London Math. Soc. 71 (1995), 197–220. · Zbl 0821.58026 [18] Sullivan, D., The density at infinity of a discrete group,Inst. Hautes Études Sci. 50 (1979), 171–202. · Zbl 0439.30034 [19] Sullivan, D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups,Acta Math. 153 (1984), 259–277. · Zbl 0566.58022
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