Skip to main content
SpringerLink
Log in

Topological and Measurable Dynamics of Lorenz Maps

  • Conference paper
Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

Abstract

We investigate the dynamics of Lorenz maps, in particular the asymptotical behaviour of the trajectory of typical points. For Lorenz maps f with negative Schwarzian derivative we give a classification of the possible metric attractors and show that either f has an ergodic absolutely continuous invariant probability measure of positive entropy or the iterates of typical points spend most of their time shadowing the trajectory of one of the two critical values. Our main tool therefore is the construction of Markov extensions for Lorenz maps which provide a unified framework to approach both the topological and the measurable aspects of the dynamics.

We study the bifurcation diagram of a smooth two parameter family of Lorenz maps which describes the parameter dependence of the kneading invariant and show that essentially every admissible kneading invariant actually occurs if the family is sufficiently rich. Finally, we adress the problem whether the kneading invariant depends monotonously on the parameters.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Bruin. Invariant measures of interval maps. PhD thesis, Technische Universiteit Delft, 1994.

    Google Scholar 

  2. H. Bruin. Combinatorics of the kneading map. Intern. J. Bif. Chaos, 5(5):1339–1349, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Bruin, G. Keller, T. Nowicki, and S. van Strien. Wild Cantor attractors exist. Annals of Mathematics, 143:97–130, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Coullet, J.-M. Gambaudo, and C. Tresser. Une nouvelle bifurcation de codimension 2: le collage de cycles. C. R. Acad. Sc. Pans Sèr. I, 299(7): 253–256, 1984.

    MathSciNet  MATH  Google Scholar 

  5. W. de Melo and S. van Strien. One-Dimensional Dynamics, volume 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin Heidelberg New York, 1993.

    Google Scholar 

  6. J.-M. Gambaudo, P. A. Glendinning, and C. Tresser. Collage de cycles et suites de Farey. C. R. Acad. Sc. Paris Sèr. I, 299(14):711–714, 1984.

    MathSciNet  MATH  Google Scholar 

  7. P. Glendinning and C. Sparrow. Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps. Physica D, 62:22–50, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Guckenheimer. A strange, strange attractor. In J. E. Marsden and M. Mc-Cracken, editors, The Hopf Bifurcation Theorem and its Applications, page 368/381. Springer, 1976.

    Google Scholar 

  9. J. Guckenheimer and R.F. Williams. Structural stability of Lorenz Attractors. Publ. Math. I.H.E.S., 50:307–320, 1979.

    Google Scholar 

  10. F. Hofbauer. On intrinsic ergodicity of piecewise monotonie transformations with positive entropy. Israel Journal of Mathematics, 34:213–237, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Hofbauer. The topological entropy of the transformation x → ax(l—x). ML Math., 90:117–141, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Hofbauer. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy ii. Israel Journal of Mathematics, 38:107–115, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Hofbauer. The structure of piecewise monotonie transformations. Erg. Th. Dyn. Sys., 1:159–178, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Hofbauer. Monotonie mod one transformations. Studia Mathematica, LXXX:17–40, 1984.

    MathSciNet  Google Scholar 

  15. F. Hofbauer. Piecewise invertible dynamical systems. Probab. Th. Rel. Fields, 72:359–386, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Hofbauer and G. Keller. Quadratic maps without asymptotic measure. Commun. Math. Phys., 127:319–337, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Homburg. Some global aspects of homoclinic bifurcations of vector fields. PhD thesis, Rijksuniversiteit Groningen, 1993.

    Google Scholar 

  18. J. H. Hubbard and C. T. Sparrow. The classification of topologically expansive Lorenz maps. Commun. Pure. Appl. Math., XLIIL 431–443, 1990.

    Article  MathSciNet  Google Scholar 

  19. G. Keller. Lifting measures to Markov extensions. Monatshefte für Mathematik, 108:183–200, 1989.

    Article  MATH  Google Scholar 

  20. G. Keller. Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems. Trans. Am. Math. Soc, 314(2):433–497, 1989.

    Article  MATH  Google Scholar 

  21. G. Keller. Exponents, attractors and Hopf decompositions for interval maps. Erg. Th. Dyn. Sys., 10:717–744, 1990.

    Article  MATH  Google Scholar 

  22. G. Keller. Zeta functions and transfer operators for piecewise monotone transformations. Commun. Math. Phys., 127:459–477, 1990.

    Article  MATH  Google Scholar 

  23. G. Keller. Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Commun. Math. Phys., 149:31–69, 1992.

    Article  MATH  Google Scholar 

  24. U. Krengel. Ergodic Theorems, volume 6 of Studies in Mathematics. de Gruyter, Berlin New York, 1985.

    Google Scholar 

  25. E. N. Lorenz. Deterministic non-periodic flow. Journal of Atmospheric Sciences, 20:130–141, 1963.

    Article  Google Scholar 

  26. M. Martens and W. de Melo. Universal models for Lorenz maps. Preprint, Oct. 1996.

    Google Scholar 

  27. J. Milnor. On the concept of attractor. Commun. Math. Phys., 99:177–195, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Milnor and W. Thurston. On iterated maps of the interval, volume 1342 of Lecture notes in Mathematics. Springer, Berlin New York, 1988.

    Google Scholar 

  29. M. Misiurewicz. Absolutely continuous measures for certain maps of an interval. Publ. Math. I.H.E.S., 53, 1981.

    Google Scholar 

  30. W. Parry. The Lorenz Attractor, volume 729 of Lecture notes in Mathematics, pages 169–187. Springer, Berlin New York, 1979.

    Google Scholar 

  31. D. Rand. The topological classification of Lorenz Attractors. Math. Proc. Camb. Phil. Soc, 83:451–460, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Sands. Topological Conditions for Positive Lyapunov Exponent in Unimodal Maps. PhD thesis, St. John’s College, Cambridge, 1994.

    Google Scholar 

  33. M. St.Pierre. Topological and Measurable Dynamics of Lorenz Maps, thesis, Friedrich-Alexander Universität, Erlangen-Nürnberg, 1998. (published in Diss. Math. CCCLXXXII, 1999).

    Google Scholar 

  34. M. Tsujii. A note on Milnor and Thurston’s monotonicity theorem. In Geometry and analysis in dynamical systems, volume 14 of Adv. Ser. Dyn. Sys., Kyoto, 1993.

    Google Scholar 

  35. M. Tsujii. A simple proof for monotonicity of entropy in the quadratic family. Erg. Th. Dyn. Sys., 20:925–934, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. F. Williams. The structure of Lorenz Attractors. Publ. Math. I.H.E.S., 50:321–347, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054, Erlangen, Germany

    Gerhard Keller & Matthias St. Pierre

Authors
  1. Gerhard Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias St. Pierre
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Freie Universität Berlin Institut für Mathematik I, Arnimallee 2-6, 14195, Berlin, Germany

    Bernold Fiedler

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Keller, G., Pierre, M.S. (2001). Topological and Measurable Dynamics of Lorenz Maps. In: Fiedler, B. (eds) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56589-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56589-2_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62524-4

  • Online ISBN: 978-3-642-56589-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation