Skip to main content
SpringerLink
Log in

Statistical Properties of Piecewise Expanding Maps of the Interval

  • Chapter
Stochastic Analysis and Applications in Physics

Part of the book series: NATO ASI Series ((ASIC,volume 449))

Abstract

We give a brief introduction to the stochastic properties of expanding maps of the interval using the transfer operator method. We also describe recent results about the statistics of entrance times and times of visits to small intervals.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
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. R. Adler, L. Flato Geodesic flows, interval maps and symbolic dynamics. Bull. Amer. Math. Soc.25229–334 (1991).

    MATH  Google Scholar 

  2. R. Bowen.Equilibrium States and the Ergodic Theory of Anosov Diffeomorphims.Lecture Notes in Mathematics470, Springer Verlag, Berlin Heidelberg New York (1975).

    Google Scholar 

  3. P. Some ergodic properties of maps of he interval. To be published in the proceedings of the summer school “Dynamical Systems & Frustrated Systems”, Temuco 1991.

    Google Scholar 

  4. Z.Coelho, P.Collet. Asymptotic limit law for close approach of two trajectories in expanding maps of the circle. Prob. Theor. and Related Topics, to appear.

    Google Scholar 

  5. P. Collet, A. Galves. Asymptotic distribution of entrance times for expanding maps of the interval. Preprint.

    Google Scholar 

  6. P.Collet, A.Galves. Statistics of close visits to the indifferent fixed point of an interval map (with A.Galves). J. Stat. Phys. (to appear).

    Google Scholar 

  7. P. Collet, S. Isola. Essential spectrum inC kof expanding Markoff maps of the interval. Commun. Math. Phys.139, 551 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  8. Z. Coelho, W. Parry. Central limit asymptotics for subshifts of finite type. Isr. J. Math.69235 (1989).

    MathSciNet  Google Scholar 

  9. N. Dunford, J.T. Schwartz.Linear Operators I.Interscience Publishers, New York (1963).

    Google Scholar 

  10. F. Hofbauer, G. Keller. Ergodic properties of invariant measures for piecewise expanding monotonic transformations. Math. Zeit.180119 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Kac. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc.531002–1010 (1947).

    Article  MATH  Google Scholar 

  12. G. Keller.Stochastic stability in some chaotic systems. Monat. Math.94, 313–333 (1982).

    Article  MATH  Google Scholar 

  13. P.L’ Ecuyer. Random number generators, good ones are hard to find. Commun of the ACM3385 (1990).

    Article  Google Scholar 

  14. F. Ledrappier.Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. and Dyn. Syst.1, 77 (1981).

    MathSciNet  MATH  Google Scholar 

  15. A. Lasota, J. Yorke. On the existence of invariant measures for piecewise mono-tonic transformationsTrans. Amer. Math. Soc. 186, 481–488, (1973).

    Article  MathSciNet  Google Scholar 

  16. D. Ruelle.Thermodynamic Formalism.Addison Wesley, Reading (1978).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre de Physique Théorique, CNRS UPR 14 Ecole Polytechnique, Palaiseau Cedex, F-91128, France

    P. Collet

Authors
  1. P. Collet
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. TELECOM Portugal, Funchal(Madeira), Portugal

    Ana Isabel Cardoso

  2. CCM, Universidade da Madeira, Funchal(Madeira), Portugal

    Margarida de Faria

  3. Fakultät für Mathematik und Informatik, Universität Mannheim, Mannheim, Germany

    Jürgen Potthoff

  4. Centre de Physique Théorique, Ecole Polytechnique, Palaiseau, France

    Roland Sénéor

  5. CCM, Universidade da Madeira, Funchal(Madeira), Portugal

    Ludwig Streit

  6. BiBoS, Universität Bielefeld, Bielefeld, Germany

    Ludwig Streit

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Collet, P. (1994). Statistical Properties of Piecewise Expanding Maps of the Interval. In: Cardoso, A.I., de Faria, M., Potthoff, J., Sénéor, R., Streit, L. (eds) Stochastic Analysis and Applications in Physics. NATO ASI Series, vol 449. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0219-3_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-0219-3_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4098-3

  • Online ISBN: 978-94-011-0219-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation