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The curvature and hyperbolicity of Hamiltonian systems

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Abstract

Curvature-type invariants of Hamiltonian systems generalize sectional curvatures of Riemannian manifolds: the negativity of the curvature is an indicator of the hyperbolic behavior of the Hamiltonian flow. In this paper, we give a self-contained description of the related constructions and facts; they lead to a natural extension of the classical results about Riemannian geodesic flows and indicate some new phenomena.

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References

  1. D. V. Anosov, Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature (Nauka, Moscow, 1967), Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 90 [Proc. Steklov Inst. Math. 90 (1967)].

    Google Scholar 

  2. V. I. Arnol’d and A. B. Givental’, “Symplectic Geometry,” in Dynamical Systems-4 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Fund. Napr. 4, pp. 5–139; Engl. transl. in Dynamical systems IV (Springer, Berlin, 1990), Encycl. Math. Sci. 4, pp. 1–136.

    Google Scholar 

  3. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, Cambridge, 1998; Faktorial, Moscow, 1999).

    Google Scholar 

  4. A. Agrachev and R. Gamkrelidze, “Symplectic Methods for Optimization and Control,” in Geometry of Feedback and Optimal Control, Ed. by B. Jakubczyk and W. Respondek (M. Dekker, New York, 1998), pp. 19–77.

    Google Scholar 

  5. A. A. Agrachev and R. V. Gamkrelidze, “Feedback-Invariant Optimal Control Theory and Differential Geometry. I: Regular Extremals,” J. Dyn. Control Syst. 3(3), 343–389 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. A. Agrachev and I. Zelenko, “Geometry of Jacobi Curves. I, II,” J. Dyn. Control Syst. 8(1), 93–140 (2002); 8 (2), 167–215 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. A. Agrachev and N. N. Shcherbakova, “Hamiltonian Systems of Negative Curvature Are Hyperbolic,” Dokl. Akad. Nauk 400(3), 295–298 (2005) [Dokl. Math. 71 (1), 49–52 (2005)].

    MathSciNet  Google Scholar 

  8. A. A. Agrachev, N. N. Chtcherbakova, and I. Zelenko, “On Curvatures and Focal Points of Dynamical Lagrangian Distributions and Their Reductions by First Integrals,” J. Dyn. Control Syst. 11(3), 297–327 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  9. W. Ballmann and M. Wojtkowski, “An Estimate for the Measure Theoretic Entropy of Geodesic Flows,” Ergodic Theory Dyn. Syst. 9(2), 271–279 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  10. F. C. Chittaro, “An Estimate for the Entropy of Hamiltonian Flows,” math.DS/0602674.

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Authors and Affiliations

  1. SISSA/ISAS, via Beirut 4, 34014, Trieste, Italy

    A. A. Agrachev

  2. Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. A. Agrachev

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  1. A. A. Agrachev
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 256, pp. 31–53.

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Agrachev, A.A. The curvature and hyperbolicity of Hamiltonian systems. Proc. Steklov Inst. Math. 256, 26–46 (2007). https://doi.org/10.1134/S0081543807010026

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  • DOI: https://doi.org/10.1134/S0081543807010026

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