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

On Hausdorff dimension and cusp excursions for Fuchsian groups

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.
Citation: Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 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.

[2]

A. F. Beardon, "The Geometry of Discrete Groups,'', Graduate Texts in Mathematics, 91 (1983).

[3]

A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra,, Acta Math., 132 (1974), 1. 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. doi: 10.1112/jlms/s1-9.2.126.

[5]

P. Billingsley, "Convergence of Probability Measures,'', Second edition, (1999).

[6]

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

[7]

K. Falconer, "Fractal Geometry,'', Mathematical Foundations and Applications, (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. 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.

[10]

I. J. Good, The fractional dimensional theory of continued fractions,, Proc. Cambridge Phil. Soc., 37 (1941), 199. 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. doi: 10.1112/S0024611598000550.

[12]

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

[13]

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

[14]

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

[15]

A. Ya. Khinchin, "Continued Fractions,'', The University of Chicago Press, (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. doi: 10.1007/BF02392046.

[18]

C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. 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. 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.

[21]

B. O. Stratmann, The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones,, in, 57 (2004), 93.

[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. 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.

[24]

D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups,, Acta Math., 153 (1984), 259. 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.

[2]

A. F. Beardon, "The Geometry of Discrete Groups,'', Graduate Texts in Mathematics, 91 (1983).

[3]

A. F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra,, Acta Math., 132 (1974), 1. 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. doi: 10.1112/jlms/s1-9.2.126.

[5]

P. Billingsley, "Convergence of Probability Measures,'', Second edition, (1999).

[6]

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

[7]

K. Falconer, "Fractal Geometry,'', Mathematical Foundations and Applications, (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. 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.

[10]

I. J. Good, The fractional dimensional theory of continued fractions,, Proc. Cambridge Phil. Soc., 37 (1941), 199. 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. doi: 10.1112/S0024611598000550.

[12]

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

[13]

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

[14]

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

[15]

A. Ya. Khinchin, "Continued Fractions,'', The University of Chicago Press, (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. doi: 10.1007/BF02392046.

[18]

C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. 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. 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.

[21]

B. O. Stratmann, The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones,, in, 57 (2004), 93.

[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. 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.

[24]

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

[1]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[2]

Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223

[3]

Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711

[4]

Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647

[5]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591

[6]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405

[7]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457

[8]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293

[9]

Nuno Luzia. Measure of full dimension for some nonconformal repellers. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 291-302. doi: 10.3934/dcds.2010.26.291

[10]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098

[11]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235

[12]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125

[13]

Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015

[14]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417

[15]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020

[16]

Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699

[17]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993

[18]

Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006

[19]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779

[20]

Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]