# American Institute of Mathematical Sciences

May  2014, 34(5): 1841-1872. doi: 10.3934/dcds.2014.34.1841

## Lyapunov spectrum for geodesic flows of rank 1 surfaces

 1 Department of Mathematics, Northwestern University, Evanston, IL 60208-2730 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil

Received  May 2012 Revised  August 2013 Published  October 2013

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.
Citation: Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
##### References:
 [1] L. M. Abramov, On the entropy of a flow,, Doklad. Acad. Nauk., 128 (1959), 873. Google Scholar [2] , D. Anosov,, private communications., (). Google Scholar [3] W. Ballmann, Axial isometries of manifolds of non-positive curvature,, Math. Annal., 259 (1982), 131. doi: 10.1007/BF01456836. Google Scholar [4] L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications,, J. Statist. Phys., 115 (2004), 1567. doi: 10.1023/B:JOSS.0000028069.64945.65. Google Scholar [5] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar [6] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. doi: 10.1007/BF01389848. Google Scholar [7] Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on no-compact spaces,, Israel. J. Math., 179 (2010), 157. doi: 10.1007/s11856-010-0076-z. Google Scholar [8] S. Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux., C. R. Math. Acad. Sci. Paris, 334 (2001), 401. doi: 10.1016/S1631-073X(02)02274-4. Google Scholar [9] P. Eberlein, Geodesic flow in certain manifolds without conjugate points,, Trans. Amer. Math. Soc., 167 (1972), 151. doi: 10.1090/S0002-9947-1972-0295387-4. Google Scholar [10] P. Eberlein, When is a geodesic flow of Anosov-type? I,II,, J. Diff. Geom., 8 (1973), 437. Google Scholar [11] P. Eberlein, Geodesic flows on negatively curved manifolds,, Trans. Amer. Math. Soc., 178 (1973), 57. doi: 10.1090/S0002-9947-1973-0314084-0. Google Scholar [12] P. Eberlein, Geometry of Nonpositively Curved Manifolds,, Chicago Lectures in Mathematics, (1996). Google Scholar [13] P. Eberlein and B. O'Neill, Visibility manifolds,, Pacific J. Math., 46 (1973), 45. doi: 10.2140/pjm.1973.46.45. Google Scholar [14] T. Fisher, Hyperbolic sets that are not locally maximal,, Ergodic Theory Dynam. Systems, 26 (2006), 1491. doi: 10.1017/S0143385706000411. Google Scholar [15] K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems,, Ergodic Theory Dynam. Systems, 29 (2009), 919. doi: 10.1017/S0143385708080462. Google Scholar [16] M. Gromov, Manifolds of negative curvature,, J. Diff. Geom., 13 (1978), 223. Google Scholar [17] S. Ito, On the topological entropy of a dynamical system,, Proc. Japan Acad., 45 (1969), 383. doi: 10.3792/pja/1195520543. Google Scholar [18] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137. Google Scholar [19] A. Katok, Entropy and closed geodesics,, Ergodic Theory Dynam. Systems, 2 (1982), 339. doi: 10.1017/S0143385700001656. Google Scholar [20] A. Katok, Nonuniform Hyperbolicity and Structure of Smooth Dynamical Systems,, (Warszawa, (1983), 1245. Google Scholar [21] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and Its Applications 54, (1995). Google Scholar [22] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds,, Ann. of Math., 148 (1998), 291. doi: 10.2307/120995. Google Scholar [23] A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy,, Ergodic Theory Dynam. Systems, 1 (1981), 451. Google Scholar [24] S. Newhouse, Entropy and volume,, Ergodic Theory Dynam. Systems, 8 (1988), 283. doi: 10.1017/S0143385700009469. Google Scholar [25] S. Newhouse, Continuity of the entropy,, Ann. of Math., 129 (1989), 215. doi: 10.2307/1971492. Google Scholar [26] T. Ohno, A weak equivalence and topological entropy,, Publ. Res. Inst. Math. Sci., 16 (1980), 289. doi: 10.2977/prims/1195187508. Google Scholar [27] Ya. Pesin, Geodesic flows in closed Riemannian manifolds without focal points,, Izv. Acad. Nauk SSSR Ser. Mat., 41 (1977), 1252. Google Scholar [28] Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, Chicago Lectures in Math., (1997). Google Scholar [29] Ya. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets,, Funct. Anal. Appl., 18 (1984), 50. Google Scholar [30] Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows,, Commun. Math. Phys., 216 (2001), 277. doi: 10.1007/s002200000329. Google Scholar [31] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics,, Sec. ed., (2004). doi: 10.1017/CBO9780511617546. Google Scholar [32] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics 79, (1981). Google Scholar

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##### References:
 [1] L. M. Abramov, On the entropy of a flow,, Doklad. Acad. Nauk., 128 (1959), 873. Google Scholar [2] , D. Anosov,, private communications., (). Google Scholar [3] W. Ballmann, Axial isometries of manifolds of non-positive curvature,, Math. Annal., 259 (1982), 131. doi: 10.1007/BF01456836. Google Scholar [4] L. Barreira and P. Doutor, Birkhoff averages for hyperbolic flows: Variational principles and applications,, J. Statist. Phys., 115 (2004), 1567. doi: 10.1023/B:JOSS.0000028069.64945.65. Google Scholar [5] R. Bowen, Topological entropy for noncompact sets,, Trans. Amer. Math. Soc., 184 (1973), 125. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar [6] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. doi: 10.1007/BF01389848. Google Scholar [7] Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on no-compact spaces,, Israel. J. Math., 179 (2010), 157. doi: 10.1007/s11856-010-0076-z. Google Scholar [8] S. Crovisier, Une remarque sur les ensembles hyperboliques localement maximaux., C. R. Math. Acad. Sci. Paris, 334 (2001), 401. doi: 10.1016/S1631-073X(02)02274-4. Google Scholar [9] P. Eberlein, Geodesic flow in certain manifolds without conjugate points,, Trans. Amer. Math. Soc., 167 (1972), 151. doi: 10.1090/S0002-9947-1972-0295387-4. Google Scholar [10] P. Eberlein, When is a geodesic flow of Anosov-type? I,II,, J. Diff. Geom., 8 (1973), 437. Google Scholar [11] P. Eberlein, Geodesic flows on negatively curved manifolds,, Trans. Amer. Math. Soc., 178 (1973), 57. doi: 10.1090/S0002-9947-1973-0314084-0. Google Scholar [12] P. Eberlein, Geometry of Nonpositively Curved Manifolds,, Chicago Lectures in Mathematics, (1996). Google Scholar [13] P. Eberlein and B. O'Neill, Visibility manifolds,, Pacific J. Math., 46 (1973), 45. doi: 10.2140/pjm.1973.46.45. Google Scholar [14] T. Fisher, Hyperbolic sets that are not locally maximal,, Ergodic Theory Dynam. Systems, 26 (2006), 1491. doi: 10.1017/S0143385706000411. Google Scholar [15] K. Gelfert and M. Rams, The Lyapunov spectrum of some parabolic systems,, Ergodic Theory Dynam. Systems, 29 (2009), 919. doi: 10.1017/S0143385708080462. Google Scholar [16] M. Gromov, Manifolds of negative curvature,, J. Diff. Geom., 13 (1978), 223. Google Scholar [17] S. Ito, On the topological entropy of a dynamical system,, Proc. Japan Acad., 45 (1969), 383. doi: 10.3792/pja/1195520543. Google Scholar [18] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Publ. Math. Inst. Hautes Études Sci., 51 (1980), 137. Google Scholar [19] A. Katok, Entropy and closed geodesics,, Ergodic Theory Dynam. Systems, 2 (1982), 339. doi: 10.1017/S0143385700001656. Google Scholar [20] A. Katok, Nonuniform Hyperbolicity and Structure of Smooth Dynamical Systems,, (Warszawa, (1983), 1245. Google Scholar [21] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and Its Applications 54, (1995). Google Scholar [22] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds,, Ann. of Math., 148 (1998), 291. doi: 10.2307/120995. Google Scholar [23] A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy,, Ergodic Theory Dynam. Systems, 1 (1981), 451. Google Scholar [24] S. Newhouse, Entropy and volume,, Ergodic Theory Dynam. Systems, 8 (1988), 283. doi: 10.1017/S0143385700009469. Google Scholar [25] S. Newhouse, Continuity of the entropy,, Ann. of Math., 129 (1989), 215. doi: 10.2307/1971492. Google Scholar [26] T. Ohno, A weak equivalence and topological entropy,, Publ. Res. Inst. Math. Sci., 16 (1980), 289. doi: 10.2977/prims/1195187508. Google Scholar [27] Ya. Pesin, Geodesic flows in closed Riemannian manifolds without focal points,, Izv. Acad. Nauk SSSR Ser. Mat., 41 (1977), 1252. Google Scholar [28] Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, Chicago Lectures in Math., (1997). Google Scholar [29] Ya. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets,, Funct. Anal. Appl., 18 (1984), 50. Google Scholar [30] Ya. Pesin and V. Sadovskaya, Multifractal analysis of conformal axiom A flows,, Commun. Math. Phys., 216 (2001), 277. doi: 10.1007/s002200000329. Google Scholar [31] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics,, Sec. ed., (2004). doi: 10.1017/CBO9780511617546. Google Scholar [32] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics 79, (1981). Google Scholar
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