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Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms
On Hausdorff dimension and cusp excursions for Fuchsian groups
1. | Fachbereich 3 - Mathematik Universitt Bremen, Postfach 33 04 40, Bibliothekstrae 1, 28359 Bremen, Germany |
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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