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1996 | 149 | 1 | 83-93
Tytuł artykułu

The box-counting dimension for geometrically finite Kleinian groups

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group. Using the 'global measure formula' for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both: the box-counting dimension and packing dimension of the limit set. Thus, by a result of Sullivan, we conclude that for a geometrically finite group these three different types of dimension coincide with the exponent of convergence of the group.
Słowa kluczowe
Rocznik
Tom
149
Numer
1
Strony
83-93
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-05-21
poprawiono
1995-08-07
Twórcy
autor
  • Department of Mathematics, University of North Texas, Denton, Texas 76203-5116, U.S.A. , urbanski@unt.edu
Bibliografia
  • [1] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-549.
  • [2] A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1-12.
  • [3] C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, preprint, Stony Brook, 1994/95.
  • [4] B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1988), 245-317.
  • [5] R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 153-170.
  • [6] E. B. Davies and N. Mandouvalos, The hyperbolic geometry and spectrum of irregular domains, Nonlinearity 3 (1990), 913-947.
  • [7] M. Denker and M. Urbański, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53-66.
  • [8] M. Denker and M. Urbański, The capacity of parabolic Julia sets, Math. Z. 211 (1992), 73-86.
  • [9] K. Falconer, Fractal Geometry, Wiley, New York, 1990.
  • [10] M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), 465-529.
  • [11] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273.
  • [12] B. Stratmann, A note on counting cuspidal excursions, Ann. Acad. Sci. Fenn. 20 (1995).
  • [13] B. Stratmann, The Hausdorff dimension of bounded geodesics on geometrically finite manifolds, accepted by Ergod. Theory Dynam. Systems; preprint in Mathematica Gottingensis 39 (1993).
  • [14] B. Stratmann and S. Velani, The Patterson measure for geometrically finite groups with parabolic elements, new and old, Proc. London Math. Soc. (3) 71 (1995), 197-220.
  • [15] D. Sullivan, The density at infinity of a discrete group, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 171-202.
  • [16] D. Sullivan, Entropy, Hausdorff measures old and new, and the limit set of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259-277.
  • [17] P. Tukia, On isomorphisms of geometrically finite Möbius groups, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985), 171-214.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv149i1p83bwm
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