Skip to main content
SpringerLink
Log in

The Metric Entropy of Endomorphisms

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove that, for a C 2 non-invertible but non-degenerate map on a compact Riemannian manifold without boundary, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents. This generalizes Ledrappier-Young’s entropy formula [5] (for negative Lyapunov exponents of diffeomorphisms) to the case of endomorphisms.

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. Barreira L., Pesin Y., Schmeling J.: Dimension and product structure of hyperbolic measures. Ann. Math. 149, 755–783 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Eckmann J.-P., Ruelle D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  3. Farmer J., Ott E., Yorke J.: The dimension of chaotic attractors. Physica 7D, 153–180 (1983)

    ADS  MathSciNet  Google Scholar 

  4. Katok, A., Strelcyn, J.-M.: Invariant Manifold, Entropy and Billiards; Smooth Maps with Singularities Lec. Not. Math. 1222, Berlin-Heidelberg-New York: Springer Verlag, 1986

  5. Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122, 509–539 (1985); The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122, 540–574 (1985)

    Google Scholar 

  6. Liu, P.-D., Qian, M.: Smooth Ergodic Theory of Random Dynamical Systems. Lecture Notes in Mathematics 1606, Berlin: Springer-Verlag, 1995

  7. Liu P.-D.: Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 240, 531–538 (2003)

    MATH  ADS  Google Scholar 

  8. Liu P.-D.: Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 284, 391–406 (2008)

    Article  MATH  ADS  Google Scholar 

  9. Mané R.: A proof of Pesin’s formula. Ergod. Th. Dynam. Syst. 1, 95–102 (1981)

    MATH  Google Scholar 

  10. Oseledeč V.-I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–221 (1968)

    Google Scholar 

  11. Ruelle D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85, 1–23 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Qian M., Zhang Z.-S.: Ergodic theory for axiom A endomorphisms. Ergod. Th. & Dynam. Sys. 15, 161–174 (1995)

    MATH  MathSciNet  Google Scholar 

  13. Qian M., Zhu S.: SRB measures and Pesin’s entropy formula for endomorphisms. Trans. Amer. Math. Soc. 354, 1453–1471 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Qian M., Xie J.-S.: Entropy formula for endomorphisms: relations between entropy, exponents and dimension. Disc. Contin. Dyn. Syst. 21, 367–392 (2008)

    MATH  MathSciNet  Google Scholar 

  15. Shu, L.: Dimension theory for invariant measures of endomorphisms. Manuscript

Download references

Author information

Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Lin Shu

Authors
  1. Lin Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lin Shu.

Additional information

Communicated by G. Gallavotti

This work is supported by National Basic Research Program of China (973 Program) (2007 CB 814800).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shu, L. The Metric Entropy of Endomorphisms. Commun. Math. Phys. 291, 491–512 (2009). https://doi.org/10.1007/s00220-009-0859-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0859-4

Keywords

Search

Navigation