The box-counting dimension for geometrically finite Kleinian groups

B. Stratmann; Mariusz Urbański

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 1, page 83-93
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Stratmann, B., and Urbański, Mariusz. "The box-counting dimension for geometrically finite Kleinian groups." Fundamenta Mathematicae 149.1 (1996): 83-93. <http://eudml.org/doc/212110>.

@article{Stratmann1996,
abstract = {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.},
author = {Stratmann, B., Urbański, Mariusz},
journal = {Fundamenta Mathematicae},
keywords = {limit sets; fractal dimensions; Patterson measure; finitely generated Fuchsian groups; exponent of convergence; Hausdorff dimension; box-counting dimension; geometrically finite Kleinian groups},
language = {eng},
number = {1},
pages = {83-93},
title = {The box-counting dimension for geometrically finite Kleinian groups},
url = {http://eudml.org/doc/212110},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Stratmann, B.
AU - Urbański, Mariusz
TI - The box-counting dimension for geometrically finite Kleinian groups
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 1
SP - 83
EP - 93
AB - 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.
LA - eng
KW - limit sets; fractal dimensions; Patterson measure; finitely generated Fuchsian groups; exponent of convergence; Hausdorff dimension; box-counting dimension; geometrically finite Kleinian groups
UR - http://eudml.org/doc/212110
ER -

References

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  2. [2] A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1-12. Zbl0277.30017
  3. [3] C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, preprint, Stony Brook, 1994/95. 
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  12. [12] B. Stratmann, A note on counting cuspidal excursions, Ann. Acad. Sci. Fenn. 20 (1995). 
  13. [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. [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. Zbl0821.58026
  15. [15] D. Sullivan, The density at infinity of a discrete group, Inst. Hautes Etudes Sci. Publ. Math. 50 (1979), 171-202. Zbl0439.30034
  16. [16] D. Sullivan, Entropy, Hausdorff measures old and new, and the limit set of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259-277. Zbl0566.58022
  17. [17] P. Tukia, On isomorphisms of geometrically finite Möbius groups, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985), 171-214. Zbl0572.30036

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