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Mosc. Math. J., 2001, Volume 1, Number 4, Pages 569–582 (Mi mmj37)  

This article is cited in 16 scientific papers (total in 16 papers)

Arithmetic coding and entropy for the positive geodesic flow on the modular surface

B. M. Gurevicha, S. R. Katokb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Department of Mathematics, Pennsylvania State University

Abstract: In this article we study geodesics on the modular surface by means of their arithmetic codes. Closed geodesics for which arithmetic and geometric codes coincide were identified in [9]. Here they are described as periodic orbits of a special flow over a topological Markov chain with countable alphabet, which we call the positive geodesic flow. We obtain an explicit formula for the ceiling function and two-sided estimates for the topological entropy of the positive geodesic flow, which turns out to be separated from one: the topological entropy of the geodesic flow on the modular surface.

Key words and phrases: Geodesic flow, modular surface, Fuchsian group, entropy, topological entropy.


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MSC: 37D40, 37B40, 20H05
Received: July 1, 2001; in revised form September 26, 2001

Citation: B. M. Gurevich, S. R. Katok, “Arithmetic coding and entropy for the positive geodesic flow on the modular surface”, Mosc. Math. J., 1:4 (2001), 569–582

Citation in format AMSBIB
\by B.~M.~Gurevich, S.~R.~Katok
\paper Arithmetic coding and entropy for the positive geodesic flow on the modular surface
\jour Mosc. Math.~J.
\yr 2001
\vol 1
\issue 4
\pages 569--582

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    This publication is cited in the following articles:
    1. S. R. Katok, I. Ugarcovici, “Geometrically Markov geodesics on the modular surface”, Mosc. Math. J., 5:1 (2005), 135–155  mathnet  mathscinet  zmath
    2. Stadlbauer M., Stratmann B.O., “Infinite ergodic theory for Kleinian groups”, Ergodic Theory and Dynamical Systems, 25:4 (2005), 1305–1323  crossref  mathscinet  zmath  isi
    3. Barreira L., Iommi G., “Suspension flows over countable Markov shifts”, Journal of Statistical Physics, 124:1 (2006), 207–230  crossref  mathscinet  zmath  adsnasa  isi
    4. Gorodnik A., “Open problems in dynamics and related fields”, Journal of Modern Dynamics, 1:1 (2007), 1–35  crossref  mathscinet  zmath  isi
    5. Kesseboehmer A., Stratmann B.O., “Homology at infinity; fractal geometry of limiting symbols for modular subgroups”, Topology, 46:5 (2007), 469–491  crossref  mathscinet  zmath  isi
    6. Katok S., Ugarcovici I., “Symbolic dynamics for the modular surface and beyond”, Bulletin of the American Mathematical Society, 44:1 (2007), 87–132  crossref  mathscinet  zmath  isi
    7. A. I. Bufetov, B. M. Gurevich, “Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials”, Sb. Math., 202:7 (2011), 935–970  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Kempton T., “Thermodynamic formalism for suspension flows over countable Markov shifts”, Nonlinearity, 24:10 (2011), 2763–2775  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Meson A.M., Vericat F., “Multifractal Analysis for the Teichmüller Flow”, Math Phys Anal Geom, 15:1 (2012), 39–60  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Katok S., Ugarcovici I., “Applications of (a, B)-Continued Fraction Transformations”, Ergod. Theory Dyn. Syst., 32:Part 2 (2012), 755–U466  crossref  mathscinet  isi
    11. Dastjerdi D.A., Lamei S., “Geodesic Flow on the Quotient Space of the Action of < Z+2,-1/Z > on the Upper Half Plane”, Analele Stiint. Univ. Ovidius C., 20:3 (2012), 37–50  mathscinet  zmath  isi
    12. Arnoux P., Schmidt T.A., “Cross Sections for Geodesic Flows and Alpha-Continued Fractions”, Nonlinearity, 26:3 (2013), 711–726  crossref  mathscinet  zmath  adsnasa  isi
    13. Iommi G., Jordan T., “Phase Transitions for Suspension Flows”, Commun. Math. Phys., 320:2 (2013), 475–498  crossref  mathscinet  zmath  isi  elib
    14. Jaerisch J., Kesseboehmer M., Lamei S., “Induced Topological Pressure for Countable State Markov Shifts”, Stoch. Dyn., 14:2 (2014), 1350016  crossref  mathscinet  zmath  isi  elib
    15. Pinsky T., “On the Topology of the Lorenz System”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 473:2205 (2017), 20170374  crossref  isi
    16. Riquelme F., “Ruelle'S Inequality in Negative Curvature”, Discret. Contin. Dyn. Syst., 38:6 (2018), 2809–2825  crossref  mathscinet  zmath  isi  scopus
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