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A Fréchet law and an Erdős–Philipp law for maximal cuspidal windings

Published online by Cambridge University Press:  18 July 2012

JOHANNES JAERISCH ,
MARC KESSEBÖHMER  and
BERND O. STRATMANN
JOHANNES JAERISCH
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan (email: jaerisch@cr.math.sci.osaka-u.ac.jp)
MARC KESSEBÖHMER
Affiliation:
Fachbereich 3—Mathematik und Informatik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany (email: mhk@math.uni-bremen.de, bos@math.uni-bremen.de)
BERND O. STRATMANN
Affiliation:
Fachbereich 3—Mathematik und Informatik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany (email: mhk@math.uni-bremen.de, bos@math.uni-bremen.de)
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Abstract

In this paper we establish a Fréchet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying extreme value theory. Subsequently, we show that this law gives rise to an Erdős–Philipp law and to various generalized Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erdős.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1008 - 1028
Copyright
Copyright © 2012 Cambridge University Press 

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References

[Aar81]Aaronson, J.. The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39 (1981), 203–234.Google Scholar
[Aar97]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[ADU93]Aaronson, J., Denker, M. and Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337(2) (1993), 495–548.Google Scholar
[Bea71]Beardon, A. F.. Inequalities for certain Fuchsian groups. Acta Math. 127 (1971), 221–258.CrossRefGoogle Scholar
[Bea95]Beardon, A. F.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91). Springer, New York, 1995; corrected reprint of the 1983 original.Google Scholar
[Ber64]Berman, S. M.. Limit theorems for the maximum term in stationary sequences. Ann. Math. Stat. 35 (1964), 502–516.Google Scholar
[BNN95]Balakrishnan, V., Nicolis, C. and Nicolis, G.. Extreme value distributions in chaotic dynamics. J. Stat. Phys. 80(1–2) (1995), 307–336.Google Scholar
[BS79]Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 1979(50) 153–170.Google Scholar
[Col01]Collet, P.. Statistics of closest return for some non-uniformly hyperbolic systems. Ergod. Th. & Dynam. Sys. 21(2) (2001), 401–420.Google Scholar
[Dav79]Davis, R. A.. Maxima and minima of stationary sequences. Ann. Probab. 7(3) (1979), 453–460.Google Scholar
[Dol97]Dolgopyat, D.. On statistical properties of geodesic flows on negatively curved surfaces. PhD Thesis, Princeton University, 1997.Google Scholar
[EP94]Epstein, D. B. A. and Petronio, C.. An exposition of Poincaré’s polyhedron theorem. Enseign. Math. (2) 40(1–2) (1994), 113–170.Google Scholar
[Fer11]Ferguson, A. J.. Dimension theory and dynamically defined sets. PhD Thesis, University of Warwick, 2011.Google Scholar
[FF08]Freitas, A. C. M. and Freitas, J. M.. On the link between dependence and independence in extreme value theory for dynamical systems. Statist. Probab. Lett. 78(9) (2008), 1088–1093.Google Scholar
[FFT10]Freitas, A. C. M., Freitas, J. M. and Todd, M.. Hitting time statistics and extreme value theory. Probab. Theory Related Fields 147(3-4) (2010), 675–710.CrossRefGoogle Scholar
[FFT11]Freitas, A. C. M., Freitas, J. M. and Todd, M.. Extreme value laws in dynamical systems for non-smooth observations. J. Stat. Phys. 142(1) (2011), 108–126.CrossRefGoogle Scholar
[Flo80]Floyd, W. J.. Group completions and limit sets of Kleinian groups. Invent. Math. 57(3) (1980), 205–218.CrossRefGoogle Scholar
[Fr{é}28]Fréchet, M.. Sur la loi de probabilité de l’écart maximum. Ann. Soc. Polon. Math. 6 (1928), 93–122.Google Scholar
[FT28]Fisher, R. A. and Tippett, L. H. C.. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge 24 (1928), 180–190.Google Scholar
[Gal72]Galambos, J.. The distribution of the largest coefficient in continued fraction expansions. Q. J. Math. Oxford Ser. (2) 23 (1972), 147–151.Google Scholar
[Gal73]Galambos, J.. The largest coefficient in continued fractions and related problems. Diophantine Approximation and its Applications (Proc. Conf., Washington, DC, 1972). Academic Press, New York, NY, 1973, pp. 101–109.Google Scholar
[GHN11]Gupta, C., Hollander, M. and Nicol, M.. Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. Ergod. Th. & Dynam. Sys. 31 (2011), 1363–1390.CrossRefGoogle Scholar
[Gne43]Gnedenko, B.. Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44 (1943), 423–453.Google Scholar
[GNO10]Gupta, C., Nicol, M. and Ott, W.. A Borel-Cantelli lemma for nonuniformly expanding dynamical systems. Nonlinearity 23(8) (2010), 1991–2008.CrossRefGoogle Scholar
[Gum58]Gumbel, E. J.. Statistics of Extremes. Columbia University Press, New York, 1958.Google Scholar
[Gup10]Gupta, C.. Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 30(3) (2010), 757–771.CrossRefGoogle Scholar
[Khi64]Khinchin, A. Y.. Continued Fractions. The University of Chicago Press, Chicago, IL, London, 1964.Google Scholar
[KS04]Kesseböhmer, M. and Stratmann, B. O.. A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dynam. Sys. 24(1) (2004), 141–170.CrossRefGoogle Scholar
[Lea74]Leadbetter, M. R.. On extreme values in stationary sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 28 (1974), 289–303.Google Scholar
[LLR83]Leadbetter, M. R., Lindgren, G. and Rootzén, H.. Extremes and Related Properties of Random Sequences and Processes (Springer Series in Statistics, 11). Springer, New York, 1983.Google Scholar
[Loy65]Loynes, R. M.. Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Stat. 36 (1965), 993–999.Google Scholar
[MFF08]Moreira Freitas, A. C. and Freitas, J. M.. Extreme values for Benedicks–Carleson quadratic maps. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1117–1133.Google Scholar
[Nic89]Nicholls, P. J.. The Ergodic Theory of Discrete Groups (London Mathematical Society Lecture Note Series, 143). Cambridge University Press, Cambridge, 1989.Google Scholar
[NN03]Nakada, H. and Natsui, R.. On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi–Perron algorithm. Monatsh. Math. 138(4) (2003), 267–288.Google Scholar
[Pat76]Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241–273.Google Scholar
[Phi76]Philipp, W.. A conjecture of Erdős on continued fractions. Acta Arith. 28(4) (1975/76), 379–386.Google Scholar
[Pol09]Pollicott, M.. Limiting distributions for geodesics excursions on the modular surface. Spectral Analysis in Geometry and Number Theory (Contemporary Mathematics, 484). American Mathematical Society, Providence, RI, 2009, pp. 177–185.Google Scholar
[Sch95]Schweiger, F.. Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.Google Scholar
[Spr69]Sprindžuk, V. G.. Mahler’s Problem in Metric Number Theory. Translated from the Russian by B. Volkmann (Translations of Mathematical Monographs, 25). American Mathematical Society, Providence, RI, 1969.Google Scholar
[SS05]Stadlbauer, M. and Stratmann, B. O.. Infinite ergodic theory for Kleinian groups. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1305–1323.CrossRefGoogle Scholar
[Sta04]Stadlbauer, M.. The return sequence of the Bowen-Series map for punctured surfaces. Fund. Math. 182(3) (2004), 221–240.Google Scholar
[Str95]Stratmann, B.. A note on counting cuspidal excursions. Ann. Acad. Sci. Fenn. Ser. A I Math. 20(2) (1995), 359–372.Google Scholar
[Sul79]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 1979(50) 171–202.Google Scholar
[Sul82a]Sullivan, D.. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. (N.S.) 6(1) (1982), 57–73.Google Scholar
[Sul82b]Sullivan, D.. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149(3-4) (1982), 215–237.Google Scholar
[Sul84]Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259–277.Google Scholar
[SV95]Stratmann, B. and Velani, S. L.. The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. Lond. Math. Soc. (3) 71(1) (1995), 197–220.CrossRefGoogle Scholar
[Tha98]Thaler, M.. The Dynkin–Lamperti arc-sine laws for measure preserving transformations. Trans. Amer. Math. Soc. 350(11) (1998), 4593–4607.Google Scholar
[Wat54]Watson, G. S.. Extreme values in samples from $m$-dependent stationary stochastic processes. Ann. Math. Stat. 25 (1954), 798–800.CrossRefGoogle Scholar