Counterexamples in non-positive curvature

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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

Abstract
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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 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," 2^nd 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," 2^nd 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.

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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," 2^nd 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," 2^nd 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.

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