January  2019, 39(1): 521-551. doi: 10.3934/dcds.2019022

Non-hyperbolic behavior of geodesic flows of rank 1 surfaces

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária - Ilha do Fundão, Av. Athos da Silveira Ramos 149, Rio de Janeiro 21945-909, Brazil

KG has been supported by CNPq (Brazil). She is very grateful for the comments by the referee

Received  April 2018 Revised  July 2018 Published  October 2018

We prove that for the geodesic flow of a rank 1 Riemannian surface which is expansive but not Anosov the Hausdorff dimension of the set of vectors with only zero Lyapunov exponents is large.

Citation: Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
References:
[1]

L. M. Abramov, On the entropy of a flow, Dokl. Akad. Nauk SSSR, 128 (1959), 873-875. 

[2]

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

[3]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin. doi: 10.1007/978-3-0348-9240-7.

[4]

R. Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations, 12 (1972), 173-179.  doi: 10.1016/0022-0396(72)90012-5.

[5]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460.  doi: 10.2307/2373793.

[6]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.

[7]

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

[8]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.

[9]

K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst., 34 (2014), 1841-1872.  doi: 10.3934/dcds.2014.34.1841.

[10]

Y. Coudène and B. Schapira, Generic measures for geodesic flows on nonpositively curved manifolds, J. Éc. polytech. Math., 1 (2014), 387-408.  doi: 10.5802/jep.14.

[11]

P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45–109, URL http://projecteuclid.org/euclid.pjm/1102946601. doi: 10.2140/pjm.1973.46.45.

[12]

P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, Ⅱ, J. Differential Geometry, 8 (1973), 437–463; ibid. 8 (1973), 565–577. doi: 10.4310/jdg/1214431801.

[13]

P. Eberlein, Geodesic flows in manifolds of nonpositive curvature, in Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), vol. 69 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2001,525–571. doi: 10.1090/pspum/069/1858545.

[14]

D.-J. Feng and P. Shmerkin, Non-conformal repellers and the continuity of pressure for matrix cocycles, Geom. Funct. Anal., 24 (2014), 1101-1128.  doi: 10.1007/s00039-014-0274-7.

[15]

K. GelfertF. Przytycki and M. Rams, On the Lyapunov spectrum for rational maps, Math. Ann., 348 (2010), 965-1004.  doi: 10.1007/s00208-010-0508-4.

[16]

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

[17]

K. Gelfert and R. O. Ruggiero, Geodesic flows modeled by expansive flows, To appear in: Proceedings of Edinburgh Mathematical Society.

[18]

M. Gerber and A. Wilkinson, Hölder regularity of horocycle foliations, J. Differential Geom., 52 (1999), 41-72.  doi: 10.4310/jdg/1214425216.

[19]

E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergodic Theory Dynam. Systems, 4 (1984), 67-80.  doi: 10.1017/S0143385700002273.

[20]

W. Hurewicz, Sur la dimension des produits cartesiens, Ann. of Math. (2), 36 (1935), 194-197.  doi: 10.2307/1968674.

[21]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[22]

A. O. LopesV. A. Rosas and R. O. Ruggiero, Cohomology and subcohomology problems for expansive, non Anosov geodesic flows, Discrete Contin. Dyn. Syst., 17 (2007), 403-422.  doi: 10.3934/dcds.2007.17.403.

[23]

M. Paternain, Expansive geodesic flows on surfaces, Ergodic Theory Dynam. Systems, 13 (1993), 153-165.  doi: 10.1017/S0143385700007264.

[24]

Y. B. Pesin and V. Sadovskaya, Multifractal analysis of conformal Axiom A flows, Comm. Math. Phys., 216 (2001), 277-312.  doi: 10.1007/s002200000329.

[25]

Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106.  doi: 10.1063/1.166242.

[26]

Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997, Contemporary views and applications. doi: 10.7208/chicago/9780226662237.001.0001.

[27]

R. O. Ruggiero, Expansive dynamics and hyperbolic geometry, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 139-172.  doi: 10.1007/BF01321305.

[28]

R. O. Ruggiero, Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.  doi: 10.1017/S0143385797060963.

[29]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[30]

L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems, 2 (1982), 109-124.  doi: 10.1017/S0143385700009615.

show all references

References:
[1]

L. M. Abramov, On the entropy of a flow, Dokl. Akad. Nauk SSSR, 128 (1959), 873-875. 

[2]

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

[3]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995, With an appendix by Misha Brin. doi: 10.1007/978-3-0348-9240-7.

[4]

R. Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations, 12 (1972), 173-179.  doi: 10.1016/0022-0396(72)90012-5.

[5]

R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460.  doi: 10.2307/2373793.

[6]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.

[7]

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

[8]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.

[9]

K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst., 34 (2014), 1841-1872.  doi: 10.3934/dcds.2014.34.1841.

[10]

Y. Coudène and B. Schapira, Generic measures for geodesic flows on nonpositively curved manifolds, J. Éc. polytech. Math., 1 (2014), 387-408.  doi: 10.5802/jep.14.

[11]

P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math., 46 (1973), 45–109, URL http://projecteuclid.org/euclid.pjm/1102946601. doi: 10.2140/pjm.1973.46.45.

[12]

P. Eberlein, When is a geodesic flow of Anosov type? Ⅰ, Ⅱ, J. Differential Geometry, 8 (1973), 437–463; ibid. 8 (1973), 565–577. doi: 10.4310/jdg/1214431801.

[13]

P. Eberlein, Geodesic flows in manifolds of nonpositive curvature, in Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), vol. 69 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2001,525–571. doi: 10.1090/pspum/069/1858545.

[14]

D.-J. Feng and P. Shmerkin, Non-conformal repellers and the continuity of pressure for matrix cocycles, Geom. Funct. Anal., 24 (2014), 1101-1128.  doi: 10.1007/s00039-014-0274-7.

[15]

K. GelfertF. Przytycki and M. Rams, On the Lyapunov spectrum for rational maps, Math. Ann., 348 (2010), 965-1004.  doi: 10.1007/s00208-010-0508-4.

[16]

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

[17]

K. Gelfert and R. O. Ruggiero, Geodesic flows modeled by expansive flows, To appear in: Proceedings of Edinburgh Mathematical Society.

[18]

M. Gerber and A. Wilkinson, Hölder regularity of horocycle foliations, J. Differential Geom., 52 (1999), 41-72.  doi: 10.4310/jdg/1214425216.

[19]

E. Ghys, Flots d'Anosov sur les 3-variétés fibrées en cercles, Ergodic Theory Dynam. Systems, 4 (1984), 67-80.  doi: 10.1017/S0143385700002273.

[20]

W. Hurewicz, Sur la dimension des produits cartesiens, Ann. of Math. (2), 36 (1935), 194-197.  doi: 10.2307/1968674.

[21]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[22]

A. O. LopesV. A. Rosas and R. O. Ruggiero, Cohomology and subcohomology problems for expansive, non Anosov geodesic flows, Discrete Contin. Dyn. Syst., 17 (2007), 403-422.  doi: 10.3934/dcds.2007.17.403.

[23]

M. Paternain, Expansive geodesic flows on surfaces, Ergodic Theory Dynam. Systems, 13 (1993), 153-165.  doi: 10.1017/S0143385700007264.

[24]

Y. B. Pesin and V. Sadovskaya, Multifractal analysis of conformal Axiom A flows, Comm. Math. Phys., 216 (2001), 277-312.  doi: 10.1007/s002200000329.

[25]

Y. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89-106.  doi: 10.1063/1.166242.

[26]

Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997, Contemporary views and applications. doi: 10.7208/chicago/9780226662237.001.0001.

[27]

R. O. Ruggiero, Expansive dynamics and hyperbolic geometry, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 139-172.  doi: 10.1007/BF01321305.

[28]

R. O. Ruggiero, Expansive geodesic flows in manifolds with no conjugate points, Ergodic Theory Dynam. Systems, 17 (1997), 211-225.  doi: 10.1017/S0143385797060963.

[29]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[30]

L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems, 2 (1982), 109-124.  doi: 10.1017/S0143385700009615.

Figure 1.  local product structure
Figure 2.  Parametrization $R = R_v$ of the local cross section in a neighborhood of a vector $v$. Here $\pi$ denotes the projection of the centre stable leaf onto the cross section given by the local product structure.
Figure 3.  Schematic construction of $\nu$: $m_\ell$-cylinders which intersect $\Sigma^\ell$ (bold), cylinders on which $\nu$ is distributed (bold blue), $\ell = 1,2,3$
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