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July  2012, 32(7): 2503-2520. doi: 10.3934/dcds.2012.32.2503

On Hausdorff dimension and cusp excursions for Fuchsian groups

Sara Munday 1,

1. 

Fachbereich 3 - Mathematik Universitt Bremen, Postfach 33 04 40, Bibliothekstrae 1, 28359 Bremen, Germany

Received  May 2011 Revised  June 2011 Published  March 2012

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarník limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This paper will describe a natural generalisation of these sets, called strict Jarník limit sets, and show how these give rise to another weak multifractal spectrum. Number-theoretical interpretations of these results in terms of continued fractions will also be given.
Keywords: Fuchsian groups, Patterson measure., Hausdorff dimension.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
References:
[1]

A. F. Beardon, The exponent of convergence of Poincaré series, Proc. London Math. Soc. (3), 18 (1968), 461-483.

[2]

A. F. Beardon, "The Geometry of Discrete Groups,'' Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.

[3]

A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., 132 (1974), 1-12. doi: 10.1007/BF02392106.

[4]

A. S. Besicovitch, Sets of fractional dimension (IV): On rational approximation to real numbers, Jour. London Math. Soc., 9 (1934), 126-131. doi: 10.1112/jlms/s1-9.2.126.

[5]

P. Billingsley, "Convergence of Probability Measures,'' Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.

[6]

C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. doi: 10.1007/BF02392718.

[7]

K. Falconer, "Fractal Geometry,'' Mathematical Foundations and Applications, John Wiley & Sons, Ldt., Chichester, 1990.

[8]

A.-H. Fan, L.-M. Liao, B.-W. Wang and J. Wu, On Khintchine exponents and Lyapunov exponents of continued fractions, Ergod. Th. Dynam. Sys., 29 (2009), 73-109. doi: 10.1017/S0143385708000138.

[9]

O. Frostman, Potential d'équilibre et capacité des ensembles avec quelque applications à la théorie des fonctions, Meddel. Lunds Univ. Math. Sem., 3 (1935), 1-118.

[10]

I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Phil. Soc., 37 (1941), 199-228. doi: 10.1017/S030500410002171X.

[11]

R. Hill and S. L. Velani, The Jarník-Besicovitch theorem for geometrically finite Kleinian groups, Proc. London Math. Soc. (3), 77 (1998), 524-550. doi: 10.1112/S0024611598000550.

[12]

J. Jaerisch and M. Kesseböhmer, The arithmetic-geometric scaling spectrum for continued fractions, Ark. Mat., 48 (2010), 335-360. doi: 10.1007/s11512-009-0102-8.

[13]

V. Jarník, Diophantische approximationen and Hausdorff mass, Mathematicheskii Sbornik, 36 (1929), 371-382.

[14]

T. Jordan and M. Rams, Increasing digits subsystems of infinite iterated function systems, Proc. of the Amer. Math. Soc., 140 (2011), 1267-1279. doi: 10.1090/S0002-9939-2011-10969-9.

[15]

A. Ya. Khinchin, "Continued Fractions,'' The University of Chicago Press, Chicago, Ill.-London, 1964.

[16]

P. J. Nicholls, "The Ergodic Theory of Discrete Groups,'' Springer-Verlag, 1983.

[17]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.

[18]

C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.

[19]

B. O. Stratmann, Fractal dimensions for Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach, Ark. Mat., 33 (1995), 385-403. doi: 10.1007/BF02559716.

[20]

B. O. Stratmann, Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements, Michigan Math. J., 46 (1999), 573-587.

[21]

B. O. Stratmann, The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones, in "Fractal Geometry and Stochastics III," Progr. Probab., 57, Birkhäuser, Basel, (2004), 93-107.

[22]

B. O. 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. doi: 10.1112/plms/s3-71.1.197.

[23]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.

[24]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379.

show all references

References:
[1]

A. F. Beardon, The exponent of convergence of Poincaré series, Proc. London Math. Soc. (3), 18 (1968), 461-483.

[2]

A. F. Beardon, "The Geometry of Discrete Groups,'' Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1983.

[3]

A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., 132 (1974), 1-12. doi: 10.1007/BF02392106.

[4]

A. S. Besicovitch, Sets of fractional dimension (IV): On rational approximation to real numbers, Jour. London Math. Soc., 9 (1934), 126-131. doi: 10.1112/jlms/s1-9.2.126.

[5]

P. Billingsley, "Convergence of Probability Measures,'' Second edition, Wiley Series in Probability and Statistics: Probability and Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.

[6]

C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. doi: 10.1007/BF02392718.

[7]

K. Falconer, "Fractal Geometry,'' Mathematical Foundations and Applications, John Wiley & Sons, Ldt., Chichester, 1990.

[8]

A.-H. Fan, L.-M. Liao, B.-W. Wang and J. Wu, On Khintchine exponents and Lyapunov exponents of continued fractions, Ergod. Th. Dynam. Sys., 29 (2009), 73-109. doi: 10.1017/S0143385708000138.

[9]

O. Frostman, Potential d'équilibre et capacité des ensembles avec quelque applications à la théorie des fonctions, Meddel. Lunds Univ. Math. Sem., 3 (1935), 1-118.

[10]

I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Phil. Soc., 37 (1941), 199-228. doi: 10.1017/S030500410002171X.

[11]

R. Hill and S. L. Velani, The Jarník-Besicovitch theorem for geometrically finite Kleinian groups, Proc. London Math. Soc. (3), 77 (1998), 524-550. doi: 10.1112/S0024611598000550.

[12]

J. Jaerisch and M. Kesseböhmer, The arithmetic-geometric scaling spectrum for continued fractions, Ark. Mat., 48 (2010), 335-360. doi: 10.1007/s11512-009-0102-8.

[13]

V. Jarník, Diophantische approximationen and Hausdorff mass, Mathematicheskii Sbornik, 36 (1929), 371-382.

[14]

T. Jordan and M. Rams, Increasing digits subsystems of infinite iterated function systems, Proc. of the Amer. Math. Soc., 140 (2011), 1267-1279. doi: 10.1090/S0002-9939-2011-10969-9.

[15]

A. Ya. Khinchin, "Continued Fractions,'' The University of Chicago Press, Chicago, Ill.-London, 1964.

[16]

P. J. Nicholls, "The Ergodic Theory of Discrete Groups,'' Springer-Verlag, 1983.

[17]

S. J. Patterson, The limit set of a Fuchsian group, Acta Math., 136 (1976), 241-273. doi: 10.1007/BF02392046.

[18]

C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.

[19]

B. O. Stratmann, Fractal dimensions for Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach, Ark. Mat., 33 (1995), 385-403. doi: 10.1007/BF02559716.

[20]

B. O. Stratmann, Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements, Michigan Math. J., 46 (1999), 573-587.

[21]

B. O. Stratmann, The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones, in "Fractal Geometry and Stochastics III," Progr. Probab., 57, Birkhäuser, Basel, (2004), 93-107.

[22]

B. O. 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. doi: 10.1112/plms/s3-71.1.197.

[23]

D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.

[24]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379.

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