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

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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

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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

show all references

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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