Skip to main content
SpringerLink
Log in

Markov extensions for multi-dimensional dynamical systems

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval can be identified with topological Markov chains with respect to measures with large entropy. We generalize this to arbitrary piecewise invertible dynamical systems under the following assumption: the total entropy of the system should be greater than the topological entropy of the boundary of some reasonable partition separating almost all orbits. We get a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy. We illustrate our results by quoting an application to a class of multi-dimensional, non-linear, non-expansive smooth dynamical systems.

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. R. Bowen,Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society153 (1971), 401–414.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Buzzi,Entropies et représentation markovienne des applications régulières de l’intervalle, Thèse, Université Paris-Sud, Orsay, 1995.

    Google Scholar 

  3. J. Buzzi,Intrinsic ergodicity of smooth interval maps, Israel Journal of Mathematics100 (1997), 125–161.

    MATH  MathSciNet  Google Scholar 

  4. J. Buzzi,Intrinsic ergodicity of affine maps on [0, 1]d, Monatshefte für Mathematik124 (1997), 97–118.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Buzzi,Entropy, volume growth and Lyapunov exponents, submitted.

  6. J. Buzzi,Ergodicité intrinsèque de produits fibrés d’applications unidimensionnelles chaotiques, Bulletin de la Société Mathématiques de France126 (1998), 51–77.

    MATH  MathSciNet  Google Scholar 

  7. J. Buzzi,Absolutely continuous invariant measures for generic piecewise affine expanding maps, International Journal of Bifurcation and Chaos, to appear.

  8. M. Denker, C. Grillenberg and K. Sigmund,Ergodic theory on compact spaces, Lecture Notes in Mathematics527, Springer-Verlag, Berlin, 1976.

    MATH  Google Scholar 

  9. B. M. Gurevič,Topological entropy of enumerable Markov chains, Soviet Mathematics Doklady10 (1969), 911–915.

    Google Scholar 

  10. B. M. Gurevič,Shift entropy and Markov measures in the path space of a denumerable graph, Soviet Mathematics Doklady11 (1970), 744–747.

    Google Scholar 

  11. F. Hofbauer,On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, I, Israel Journal of Mathematics34 (1979), 213–237;II,38 (1981), 107–115.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. Hofbauer,Piecewise invertible dynamical systems, Probability Theory and Related Fields72 (1986), 359–386.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Katok,Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l’Institut des Hautes Études Scientifiques51 (1980), 137–173.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Keller,Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Transactions of the American Mathematical Society314 (1989), 433–497.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Keller,Lifting measures to Markov extensions, Monatshefte für Mathematik108 (1989), 183–200.

    Article  Google Scholar 

  16. S. E. Newhouse,On some results of Hofbauer on maps on the interval, Proceedings of the conferenceDynamical systems and related topics held in Nagoya in 1990 and published as volume 9 of theAdvanced Series in Dynamical Systems, World Science Publishing, River Edge, NJ, 1991, pp. 407–421.

    Google Scholar 

  17. D. Newton and W. Parry,On a factor automorphism of a normal dynamical system, Annals of Mathematical Statistics37 (1966), 1528–1533.

    MathSciNet  Google Scholar 

  18. W. Parry,Entropy and Generators in Ergodic Theory, W. A. Benjamin, New York, 1969.

    MATH  Google Scholar 

  19. D. J. Rudolph,Fundamentals of Measurable Dynamics, Clarendon Press, Oxford, 1990.

    MATH  Google Scholar 

  20. B. Weiss,Intrinsically ergodic systems, Bulletin of the American Mathematical Society76 (1970), 1266–1269.

    MATH  Google Scholar 

  21. Y. Yomdin,Volume growth and entropy, Israel Journal of Mathematics57 (1987), 285–300.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Topologie, Université de Bourgogne / CNRS, B.P. 47870, 21078, Dijon Cedex, France

    Jérôme Buzzi

Authors
  1. Jérôme Buzzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jérôme Buzzi.

Additional information

Part of this work was done at Université Paris-Sud, dép. de mathématiques, Orsay.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buzzi, J. Markov extensions for multi-dimensional dynamical systems. Isr. J. Math. 112, 357–380 (1999). https://doi.org/10.1007/BF02773488

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773488

Keywords

Search

Navigation