November  2011, 30(4): 1095-1106. doi: 10.3934/dcds.2011.30.1095

Counterexamples in non-positive curvature

1. 

Université de Bretagne Occidentale, 6 av. Le Gorgeu, 29238 Brest cedex, France

2. 

LAMFA, Université Picardie Jules Verne, 33 rue St Leu 80000 Amiens, France

Received  April 2010 Revised  August 2010 Published  May 2011

We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.
Citation: Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095
References:
[1]

D. V. Anosov, Geodesic flows on closed riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967).

[2]

W. Ballmann, M. Brin and R. Spatzier, Structure of manifolds of nonpositive curvature. II, Ann. of Math., 122 (1985), 205-235. doi: 10.2307/1971303.

[3]

P. Billingsley, Convergence of probability measures, "Wiley Series in Probability and Statistics: Probability and Statistics," 2nd edition, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.

[4]

Yu. D. Burago and S. Z. Shefel, The geometry of surfaces in Euclidean spaces, "Geometry, III," Encyclopaedia Math. Sci., 48, Springer, Berlin, (1992), 1-85, 251-256.

[5]

Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on non-compact spaces, Israel J. Math., 179 (2010), 157-172. doi: 10.1007/s11856-010-0076-z.

[6]

P. Eberlein, Geodesic flows on negatively curved manifolds I, Ann. Math. II Ser., 95 (1972), 492-510. doi: 10.2307/1970869.

[7]

P. Eberlein, "Geometry of Nonpositively Curved Manifolds," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, vii+449.

[8]

J. Hadamard, Les surfaces courbures opposées et leurs lignes géodésiques, dans Oeuvres (1898), 2, 729-775, Paris: Editions du Centre National de la Recherche Scientifique, (1968), 2296.

[9]

G. Knieper, Hyperbolic dynamics and Riemannian geometry, Handbook of Dynamical Systems, 1A (2002), 453-545.

[10]

G. Link, M. Peigné and J. C. Picaud, Sur les surfaces non-compactes de rang un, L'enseignement Mathématique, 52 (2006), 3-36.

[11]

C. Robinson, Dynamical systems. Stability, symbolic dynamics, and chaos, "Studies in Advanced Mathematics," 2nd edition, CRC Press, Boca Raton, FL, 1999.

[12]

K. Sigmund, On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37. doi: 10.2307/2373591.

show all references

References:
[1]

D. V. Anosov, Geodesic flows on closed riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967).

[2]

W. Ballmann, M. Brin and R. Spatzier, Structure of manifolds of nonpositive curvature. II, Ann. of Math., 122 (1985), 205-235. doi: 10.2307/1971303.

[3]

P. Billingsley, Convergence of probability measures, "Wiley Series in Probability and Statistics: Probability and Statistics," 2nd edition, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1999.

[4]

Yu. D. Burago and S. Z. Shefel, The geometry of surfaces in Euclidean spaces, "Geometry, III," Encyclopaedia Math. Sci., 48, Springer, Berlin, (1992), 1-85, 251-256.

[5]

Y. Coudene and B. Schapira, Generic measures for hyperbolic flows on non-compact spaces, Israel J. Math., 179 (2010), 157-172. doi: 10.1007/s11856-010-0076-z.

[6]

P. Eberlein, Geodesic flows on negatively curved manifolds I, Ann. Math. II Ser., 95 (1972), 492-510. doi: 10.2307/1970869.

[7]

P. Eberlein, "Geometry of Nonpositively Curved Manifolds," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, vii+449.

[8]

J. Hadamard, Les surfaces courbures opposées et leurs lignes géodésiques, dans Oeuvres (1898), 2, 729-775, Paris: Editions du Centre National de la Recherche Scientifique, (1968), 2296.

[9]

G. Knieper, Hyperbolic dynamics and Riemannian geometry, Handbook of Dynamical Systems, 1A (2002), 453-545.

[10]

G. Link, M. Peigné and J. C. Picaud, Sur les surfaces non-compactes de rang un, L'enseignement Mathématique, 52 (2006), 3-36.

[11]

C. Robinson, Dynamical systems. Stability, symbolic dynamics, and chaos, "Studies in Advanced Mathematics," 2nd edition, CRC Press, Boca Raton, FL, 1999.

[12]

K. Sigmund, On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37. doi: 10.2307/2373591.

[1]

Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933

[2]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148

[3]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

[4]

Yutian Lei. On the integral systems with negative exponents. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1039-1057. doi: 10.3934/dcds.2015.35.1039

[5]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158

[6]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581

[7]

Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153

[8]

Gianni Di Pillo, Giampaolo Liuzzi, Stefano Lucidi. A primal-dual algorithm for nonlinear programming exploiting negative curvature directions. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 509-528. doi: 10.3934/naco.2011.1.509

[9]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173

[10]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[11]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[12]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390

[13]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[14]

Dubi Kelmer, Hee Oh. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds. Journal of Modern Dynamics, 2021, 17: 401-434. doi: 10.3934/jmd.2021014

[15]

Gabriela P. Ovando. The geodesic flow on nilpotent Lie groups of steps two and three. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 327-352. doi: 10.3934/dcds.2021119

[16]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Ghost effect by curvature in planar Couette flow. Kinetic and Related Models, 2011, 4 (1) : 109-138. doi: 10.3934/krm.2011.4.109

[17]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441

[18]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153

[19]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016

[20]

Lixia Yuan, Wei Zhao. On a curvature flow in a band domain with unbounded boundary slopes. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 261-283. doi: 10.3934/dcds.2021115

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]