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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

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