American Institute of Mathematical Sciences

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

Counterexamples in non-positive curvature

Yves Coudène 1, and  Barbara Schapira 2,

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.
Keywords: Dynamical systems, geodesic flow, negative curvature..
Mathematics Subject Classification: Primary: 37B10, 37C40; Secondary: 34C2.
Citation: Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete & 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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[1]

Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete & 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 & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[3]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119
[4]

Yutian Lei. On the integral systems with negative exponents. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[14]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Ghost effect by curvature in planar Couette flow. Kinetic & Related Models, 2011, 4 (1) : 109-138. doi: 10.3934/krm.2011.4.109
[15]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[16]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153
[17]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016
[18]

David L. Finn. Noncompact manifolds with constant negative scalar curvature and singular solutions to semihnear elliptic equations. Conference Publications, 1998, 1998 (Special) : 262-275. doi: 10.3934/proc.1998.1998.262
[19]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[20]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences