Israel Journal of Mathematics

, Volume 225, Issue 2, pp 609–659 | Cite as

Entropy in the cusp and phase transitions for geodesic flows

  • Godofredo Iommi
  • Felipe Riquelme
  • Anibal Velozo


In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one hand on symbolic methods and Markov partitions, and on the other on geometric techniques and approximation properties at the level of groups.


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  1. [Abr59]
    L. M. Abramov, On the entropy of a flow, Doklady Akademii Nauk SSSR 128 (1959), 873–875.MathSciNetzbMATHGoogle Scholar
  2. [AK42]
    W. Ambrose and S. Kakutani, Structure and continuity of measurable flows, Duke Mathematical Journal 9 (1942), 25–42.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BG11]
    A. I. Bufetov and B. M. Gurevich, Existence and uniqueness of a measure with maximal entropy for the Teichmüller flow on the moduli space of abelian differentials, Rossiĭskaya Akademiya Nauk. Matematicheskiĭ Sbornik 202 (2011), 3–42.CrossRefzbMATHGoogle Scholar
  4. [BI06]
    L. Barreira and G. Iommi, Suspension flows over countable Markov shifts, Journal of Statistical Physics 124 (2006), 207–230.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Bow73]
    R. Bowen, Symbolic dynamics for hyperbolic flows, American Journal of Mathematics 95 (1973), 429–460.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BR75]
    R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Inventiones Mathematicae 29 (1975), 181–202.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BRW04]
    L. Barreira, L. Radu and C. Wolf, Dimension of measures for suspension flows, Dynamical Systems 19 (2004), 89–107.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [BS03]
    J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory and Dynamical Systems 23 (2003), 1383–1400.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [BS81a]
    L. A. Bunimovich and Y. G. Sinaĭ, Markov partitions for dispersed billiards, Communications in Mathematical Physics 78 (1980/81), 247–280.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [BS81b]
    L. A. Bunimovich and Y. G. Sinaĭ, Statistical properties of Lorentz gas with periodic configuration of scatterers, Communications in Mathematical Physics 78 (1980/81), 479–497.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Buz10]
    J. Buzzi, Puzzles of quasi-finite type, zeta functions and symbolic dynamics for multi-dimensional maps, Université de Grenoble. Annales de l’Institut Fourier 60 (2010), 801–852.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Cou03]
    Y. Coudene, Gibbs measures on negatively curved manifolds, Journal of Dynamical and Control Systems 9 (2003), 89–101.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Dao13]
    Y. Daon, Bernoullicity of equilibrium measures on countable Markov shifts, Discrete and Continuous Dynamical Systems 33 (2013), 4003–4015.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [DGR11]
    L. J. Díaz, K. Gelfert and M. Rams, Rich phase transitions in step skew products, Nonlinearity 24 (2011), 3391–3412.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [DOP00]
    F. Dal’bo, J.-P. Otal and M. Peigné, Séries de Poincaré des groupes géométriquement finis, Israel Journal of Mathematics 118 (2000), 109–124.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [DP98]
    F. Dal’bo and M. Peigné, Some negatively curved manifolds with cusps, mixing and counting, Journal für die Reine und Angewandte Mathematik 497 (1998), 141–169.MathSciNetzbMATHGoogle Scholar
  17. [DT]
    N. Dobbs and M. Todd, Free energy jumps up, preprint, arXiv:1512.09245..Google Scholar
  18. [EKP15]
    M. Einsiedler, S. Kadyrov and A. Pohl, Escape of mass and entropy for diagonal flows in real rank one situations, Israel Journal of Mathematics 210 (2015), 245–295.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [ELMV12]
    M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh, The distribution of closed geodesics on the modular surface, and Duke’s theorem, Enseignement des Mathématiques 58 (2012), 249–313.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [FJLR15]
    A.-H. Fan, T. Jordan, L. Liao and M. Rams, Multifractal analysis for expanding interval maps with infinitely many branches, Transactions of the American Mathematical Society 367 (2015), 1847–1870.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Gur69]
    B. M. Gurevič, Topological entropy of a countable Markov chain, Soviet Mathematics. Doklady 10 (1969), 911–915.Google Scholar
  22. [Ham]
    U. Hamenstädt, Symbolic dynamics for the teichmüller flow, preprint, arXiv:1112.6107.Google Scholar
  23. [HIH77]
    E. Heintze and H.-C. Im Hof, Geometry of horospheres, Journal of Differential Geometry 12 (1977), 481–491 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  24. [IJ13]
    G. Iommi and T. Jordan, Phase transitions for suspension flows, Communications in Mathematical Physics 320 (2013), 475–498.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [IJT15]
    G. Iommi, T. Jordan and M. Todd, Recurrence and transience for suspension flows, Israel Journal of Mathematics 209 (2015), 547–592.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [IT10]
    G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Communications in Mathematical Physics 300 (2010), 65–94.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [IT13]
    G. Iommi and M. Todd, Transience in dynamical systems, Ergodic Theory and Dynamical Systems 33 (2013), 1450–1476.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [JKL14]
    J. Jaerisch, M. Kesseböhmer and S. Lamei, Induced topological pressure for countable state Markov shifts, Stochastics and Dynamics 14 (2014), 1350016, 31 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Kem11]
    T. Kempton, Thermodynamic formalism for suspension flows over countable Markov shifts, Nonlinearity 24 (2011), 2763–2775.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [KKLM17]
    S. Kadyrov, D. Kleinbock, E. Lindenstrauss and G. A. Margulis, Singular systems of linear forms and non-escape of mass in the space of lattices, Journal d’Analayse Mathématique 133 (2017), 253–277.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [MU03]
    R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, Vol. 148, Cambridge University Press, Cambridge, 2003.Google Scholar
  32. [OP04]
    J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Mathematical Journal 125 (2004), 15–44.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Pat76]
    S. J. Patterson, The limit set of a Fuchsian group, Acta Mathematica 136 (1976), 241–273.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [PP90]
    W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187/188 (1990).Google Scholar
  35. [PPS15]
    F. Paulin, M. Pollicott and B. Schapira, Equilibrium states in negative curvature, Astérisque 373 (2015).Google Scholar
  36. [PRL]
    F. Przytycki and J. Rivera-Letelier, Geometric pressure for multimodal maps of the interval, Memoirs of the American Mathematical Society, to appear.Google Scholar
  37. [Rat73]
    M. Ratner, Markov partitions for Anosov flows on n-dimensional manifolds, Israel Journal of Mathematics 15 (1973), 92–114.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [RV]
    F. Riquelme and A. Velozo, Escape of mass and entropy for geodesic flows, Ergodic Theory and Dynamical Systems, to appear, arXiv:1610.04683.Google Scholar
  39. [Sar99]
    O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory and Dynamical Systems 19 (1999), 1565–1593.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [Sar01]
    O. Sarig, Phase transitions for countable Markov shifts, Communications in Mathematical Physics 217 (2001), 555–577.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Sar03]
    O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proceedings of the American Mathematical Society 131 (2003), 1751–1758.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [Sar15]
    O. Sarig, Thermodynamic formalism for countable Markov shifts, in Hyperbolic Dynamics, Fluctuations and Large Deviations, Proceedings of Symposia in Pure Mathematics, Vol. 89, American Mathematical Society, Providence, RI, 2015, pp. 81–117.CrossRefGoogle Scholar
  43. [Sav98]
    S. V. Savchenko, Special flows constructed from countable topological Markov chains, Funktsional’nyĭ Analiz i ego Prilozheniya 32 (1998), 40–53, 96.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [Sin72]
    J. G. Sinaĭ, Gibbs measures in ergodic theory, Akademija Nauk SSSR i Moskovskoe Matematičeskoe Obščestvo. Uspehi Matematičeskih Nauk 27 (1972), 21–64.MathSciNetzbMATHGoogle Scholar
  45. [Sul84]
    D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Mathematica 153 (1984), 259–277.zbMATHGoogle Scholar
  46. [Wal82]
    P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York–Berlin, 1982.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Godofredo Iommi
    • 1
  • Felipe Riquelme
    • 2
  • Anibal Velozo
    • 3
  1. 1.Facultad de MatemáticasPontificia Universidad Católica de Chile (PUC)SantiagoChile
  2. 2.Instituto de MatemáticasPontificia Universidad Católica de ValparaísoCerro Barón, ValparaísoChile
  3. 3.Princeton UniversityPrincetonUSA

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