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Lyapunov spectrum for geodesic flows of rank 1 surfaces

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  • We give estimates on the Hausdorff dimension of the levels sets of the Lyapunov exponent for the geodesic flow of a compact rank 1 surface. We show that the level sets of points with small (but non-zero) exponents has full Hausdorff dimension, but carries small topological entropy.
    Mathematics Subject Classification: Primary: 37D25, 37D35, 28D20, 37C45.


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  • [1]

    L. M. Abramov, On the entropy of a flow, Doklad. Acad. Nauk., 128 (1959), 873-875.


    W. Ballmann, Axial isometries of manifolds of non-positive curvature, Math. Annal., 259 (1982), 131-144.doi: 10.1007/BF01456836.


    L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications, J. Statist. Phys., 115 (2004), 1567-1603.doi: 10.1023/B:JOSS.0000028069.64945.65.


    R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.doi: 10.1090/S0002-9947-1973-0338317-X.


    R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.doi: 10.1007/BF01389848.


    Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on no-compact spaces, Israel. J. Math., 179 (2010), 157-172.doi: 10.1007/s11856-010-0076-z.


    S. Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux. C. R. Math. Acad. Sci. Paris, 334 (2001), 401-404.doi: 10.1016/S1631-073X(02)02274-4.


    P. Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc., 167 (1972), 151-170.doi: 10.1090/S0002-9947-1972-0295387-4.


    P. Eberlein, When is a geodesic flow of Anosov-type? I,II, J. Diff. Geom., 8 (1973), 437-463.


    P. Eberlein, Geodesic flows on negatively curved manifolds, Trans. Amer. Math. Soc., 178 (1973), 57-82.doi: 10.1090/S0002-9947-1973-0314084-0.


    P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.


    P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45-109.doi: 10.2140/pjm.1973.46.45.


    T. Fisher, Hyperbolic sets that are not locally maximal, Ergodic Theory Dynam. Systems, 26 (2006), 1491-1509.doi: 10.1017/S0143385706000411.


    K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems, Ergodic Theory Dynam. Systems, 29 (2009), 919-940.doi: 10.1017/S0143385708080462.


    M. Gromov, Manifolds of negative curvature, J. Diff. Geom., 13 (1978), 223-230.


    S. Ito, On the topological entropy of a dynamical system, Proc. Japan Acad., 45 (1969), 383-840.doi: 10.3792/pja/1195520543.


    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137-173.


    A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems, 2 (1982), 339-365.doi: 10.1017/S0143385700001656.


    A. Katok, Nonuniform Hyperbolicity and Structure of Smooth Dynamical Systems, (Warszawa, 1983) (Proceedings of the International Congress of Mathematicians). Eds. Z. Ciesielski and C. Olech, North-Holland, 1984, pp. 1245-1253.


    A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press, Cambridge, 1995.


    G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds, Ann. of Math., 148 (1998), 291-314.doi: 10.2307/120995.


    A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergodic Theory Dynam. Systems, 1 (1981), 451-459.


    S. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8 (1988), 283-299.doi: 10.1017/S0143385700009469.


    S. Newhouse, Continuity of the entropy, Ann. of Math., 129 (1989), 215-235.doi: 10.2307/1971492.


    T. Ohno, A weak equivalence and topological entropy, Publ. Res. Inst. Math. Sci., 16 (1980), 289-298.doi: 10.2977/prims/1195187508.


    Ya. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Acad. Nauk SSSR Ser. Mat., 41 (1977), 1252-1288.


    Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1997.


    Ya. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl., 18 (1984), 50-63.


    Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows, Commun. Math. Phys., 216 (2001), 277-312.doi: 10.1007/s002200000329.


    D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, Sec. ed., Cambridge University Press, Cambridge, 2004.doi: 10.1017/CBO9780511617546.


    P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer, New York, 1981.

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