Skip to main content

Lyapunov exponents and complexity for interval maps

Chapter 4: Deterministic Dynamical Systems

Part of the Lecture Notes in Mathematics book series (LNM,volume 1486)

Keywords

  • Periodic Orbit
  • Lyapunov Exponent
  • Invariant Measure
  • General Dynamical System
  • Solenoidal Attractor

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   52.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.M. Blokh, M.Yu. Lyubich, Measurable dynamics of S-unimodal maps of the interval, Preprint (1990), Stony Brook.

    Google Scholar 

  2. A.A. Brudno, Entropy and the complexity of the trajectories of a dynamical system. Trans. Moscow Math, Soc. 44 (1982), 127–151.

    MathSciNet  MATH  Google Scholar 

  3. P. Grassberger, Complexity and forecasting in dynamical systems, in: Measures of Complexity, Proceedings Rome, 1987, Lectures Notes in Physics 314.

    Google Scholar 

  4. R.M. Gray, Probability, Random Processes, and Ergodic Properties, Springer-Verlag, 1988.

    Google Scholar 

  5. J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys. 70 (1979), 133–160.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. F. Hofbauer, G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Zeitschrift 180 (1982), 119–140.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. F. Hofbauer, G. Keller, Quadratic maps without asymptotic measure, Commun. Math. Phys. 127 (1990), 319–337.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. S. Johnson, Singular measures without restrictive intervals, Commun. Math. Phys. 110 (1987), 185–190.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. G. Keller, Markkov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc. 314 (1989), 433–497.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. G. Keller, Exponents, attractors, and Hopf decompositions for interval maps, Preprint, 1988, (to appear in Ergod. Th. & Dynam. Sys.)

    Google Scholar 

  11. A.N. Kolmogorov, Three approaches to the definition of the concept of the “amount of information”, Selected Transl. Math. Statist. and Probab. 7 (1968), 293–302.

    MathSciNet  MATH  Google Scholar 

  12. F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval. Ergod. Th.& Dynam. Sys. 1 (1981), 77–93.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. M. Martens, Interval Dynamics, Thesis (1990), Delft.

    Google Scholar 

  14. M. Misiurewicz, Absolutely continuous invariant measures for certain maps of an interval, Publ. Math. I.H.E.S. 53 (1981), 17–51.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32 (1977), 54–114.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. D. Ruelle, An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9 (1978), 83–87.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260–267.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Keller, G. (1991). Lyapunov exponents and complexity for interval maps. In: Arnold, L., Crauel, H., Eckmann, JP. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086671

Download citation

  • DOI: https://doi.org/10.1007/BFb0086671

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54662-7

  • Online ISBN: 978-3-540-46431-0

  • eBook Packages: Springer Book Archive