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.

[2]

, D. Anosov,, private communications., ().

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[12]

P. Eberlein, Geometry of Nonpositively Curved Manifolds,, Chicago Lectures in Mathematics, (1996).

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

A. Katok, Nonuniform Hyperbolicity and Structure of Smooth Dynamical Systems,, (Warszawa, (1983), 1245.

[21]

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

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, Chicago Lectures in Math., (1997).

[29]

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

[30]

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

[31]

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

[32]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics 79, (1981).

show all references

References:
[1]

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

[2]

, D. Anosov,, private communications., ().

[3]

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

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[12]

P. Eberlein, Geometry of Nonpositively Curved Manifolds,, Chicago Lectures in Mathematics, (1996).

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

A. Katok, Nonuniform Hyperbolicity and Structure of Smooth Dynamical Systems,, (Warszawa, (1983), 1245.

[21]

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

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications,, Chicago Lectures in Math., (1997).

[29]

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

[30]

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

[31]

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

[32]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics 79, (1981).

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