Skip to main content
SpringerLink
Log in

Lyapunov exponents and invariant measures on a projective bundle

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

A discrete dynamical system generated by a diffeomorphism f on a compact manifold is considered. The Morse spectrum is the limit set of Lyapunov exponents of periodic pseudotrajectories. It is proved that the Morse spectrum coincides with the set of averagings of the function ϕ(x, e) = ln|Df(x)e| over the invariant measures of the mapping induced by the differential Df on the projective bundle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. M. Lyapunov, The General Problem of the Stability of Motion (Gostekhizdat, Moscow–Leningrad, 1950; Taylor & Francis, Ltd., London, 1992).

    MATH  Google Scholar 

  2. C. Conley, Isolated Invariant Set and theMorse Index, in CBMS Regional Conf. Ser. (Amer.Math. Soc., Providence, RI, 1978), Vol.38.

    Google Scholar 

  3. M. Shub, Stabilitéglobale de systèmes dynamiques, in Asterisque (Soc. Math. France, Paris, 1978), Vol.56.

  4. F. Colonius and W. Kliemann, The Dynamics of Control (Birkhôuser Boston, Boston, MA, 2000).

    Book  MATH  Google Scholar 

  5. G. Osipenko, “Dynamical Systems, Graphs, and Algorithms,” in Lectures Notes inMath. (Springer-Verlag, Berlin, 2007), Vol. 1889.

  6. G. S. Osipenko, “Spectrumof a dynamical system and applied symbolic dynamics,” J. Math. Anal. Appl. 252 (2), 587–616 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. S. Osipenko, J. V. Romanovsky, N. B. Ampilova and E. I. Petrenko, “Computation of the Morse spectrum,” J.Math. Sci. (N. Y.) 120 (2), 1155–1166 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge Univ. Press, Cambridge, 1995).

    Book  MATH  Google Scholar 

  9. C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics and Chaos (CRC Press, Boca Raton, FL, 1995).

    MATH  Google Scholar 

  10. C. S. Hsu, Cell-to-Cell Mapping. A Method of Global Analysis for Nonlinear Systems, in Appl.Math. Sci. (Springer, New York, 1987), Vol.64.

  11. G. S. Osipenko, “On the symbolic image of a dynamical system,” in: Boundary-Value Problems (Perm. Politekh. Inst., Perm’, 1983), pp. 101–105 [in Russian].

    Google Scholar 

  12. G. S. Osipenko, “Localization of the chain recurrent set by symbolic dynamics methods,” in Proceedings of Dynamic Systems and Applications, Vol. 1 (Dynamic, Atlanta, GA, 1994), pp. 227–282.

    Google Scholar 

  13. G. Osipenko, “Symbolic images and invariant measures of dynamical systems,” Ergodic Theory Dynam. Systems 30 (4), 1217–1237 (2010).

    Article  MathSciNet  Google Scholar 

  14. A. Katok and B. Hasselblat, Introduction to theModern Theory of Dynamical Systems, in Encyclopedia Math. Appl. (Cambridge Univ. Press, Cambridge, 1995), Vol. 54.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LomonosovMoscow State University, Moscow, Russia

    G. S. Osipenko

Authors
  1. G. S. Osipenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. S. Osipenko.

Additional information

Original Russian Text © G. S. Osipenko, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 4, pp. 549–561.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Osipenko, G.S. Lyapunov exponents and invariant measures on a projective bundle. Math Notes 101, 666–676 (2017). https://doi.org/10.1134/S0001434617030245

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434617030245

Keywords

Search

Navigation