Skip to main content
SpringerLink
Log in

Horseshoes for Anosov Systems on Fibers Driven by an Equicontinuous System

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we construct random horseshoes of Anosov systems driven by an equicontinuous system based on an ergodic measure with positive entropy.

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. Adler, R. L., Weiss, B.: Entropy, a complete metric invariant for automorphisms of the torus. Proc. Natl. Acad. Sci. USA, 57, 1573–1576 (1967)

    Article  MathSciNet  Google Scholar 

  2. Adler, R. L., Weiss, B.: Similarity of automorphisms of the torus. Mem. Amer. Math. Soc., No. 98, American Mathematical Society, Providence, RI, ii+43 pp., 1970

    MATH  Google Scholar 

  3. Anosov, D.: Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math., No. 90, American Mathematical Society, Providence, RI, iv+235 pp., 1969

    MATH  Google Scholar 

  4. Berg, K.: On the conjugacy problem for K-systems, Ph.D. Thesis, University of Minnesota, 1967

  5. Bowen, R.: Markov partitions for Axiom A diffeomorphisms. Amer. J. Math., 92, 725–747 (1970)

    Article  MathSciNet  Google Scholar 

  6. Bowen, R.: Markov partitions are not smooth. Proc. Amer. Math. Soc., 71, 130–132 (1970)

    Article  MathSciNet  Google Scholar 

  7. Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math., 29, 181–202 (1975)

    Article  MathSciNet  Google Scholar 

  8. Bowen, R.: On Axiom A Diffeomorphisms. CBMS Reg. Conf. Ser. Math., No. 35, American Mathematical Society, Providence, RI, vii+45 pp., 1978

    MATH  Google Scholar 

  9. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Second Eevised Edition. With a preface by David Ruelle. Lecture Notes in Math., Vol. 470, Springer-Verlag, Berlin, viii+75 pp., 2008

    Book  Google Scholar 

  10. Huang, W., Ye, X.: A local variational relation and applications. Israel J. Math., 151, 237–279 (2006)

    Article  MathSciNet  Google Scholar 

  11. Huang, W., Lu, K.: Entropy, chaos and weak horseshoe for infinite dimensional random dynamical systems. Comm. Pure Appl. Math., 70(10), 1987–2036 (2017)

    Article  MathSciNet  Google Scholar 

  12. Huang, W., Lian, Z., Lu, K.: Ergodic theory of random Anosov systems mixing on fibers, arXiv:1612.08394v2.

  13. Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci., 51, 137–173 (1980)

    Article  MathSciNet  Google Scholar 

  14. Kerr, D., Li, H.: Independence in topological and C*-dynamics. Math. Ann., 338(4), 869–926 (2007)

    Article  MathSciNet  Google Scholar 

  15. Kolmogorov, A. N.: A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces (Russian). Dokl. Akad. Nauk SSSR (N.S.), 119, 861–864 (1958)

    MathSciNet  MATH  Google Scholar 

  16. Lian, Z., Young, L. S.: Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert Spaces. Ann. Henri Poincaré, 12, 1081–1108 (2011)

    Article  MathSciNet  Google Scholar 

  17. Lian, Z., Young, L. S.: Lyapunov exponents, periodic orbits and horseshoes for semiflows on Hilbert Spaces. J. Amer. Math. Soc., 25(3), 637–665 (2012)

    Article  MathSciNet  Google Scholar 

  18. Ruelle, D.: A measure associated with Axiom A attractors. Amer. J. Math., 98, 619–654 (1976)

    Article  MathSciNet  Google Scholar 

  19. Shub, M.: Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987

    Book  Google Scholar 

  20. Sinai, Ya. G.: On the concept of entropy for a dynamic system (Russian). Dokl. Akad. Nauk SSSR, 124, 768–771 (1959)

    MathSciNet  MATH  Google Scholar 

  21. Sinai, Ya. G.: Markov partitions and C-diffeomorphisms. Funct. Anal. Appl., 2, 61–82 (1968)

    Article  MathSciNet  Google Scholar 

  22. Sinai, Ya. G.: Construction of Markov partitions. Funct. Anal. Appl., 2, 245–253 (1968)

    Article  Google Scholar 

  23. Sinai, Ya. G.: Gibbs measure in ergodic theory. Russian Math. Surveys, 27, 21–69 (1972)

    Article  MathSciNet  Google Scholar 

  24. Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc., 73, 747–817 (1967)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and valuable comments.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China

    Wen Huang

  2. College of Mathematical Sciences, Sichuan University, Chengdu, 610016, P. R. China

    Zeng Lian

Authors
  1. Wen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zeng Lian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zeng Lian.

Additional information

Wen Huang is partially supported by NNSF of China (Grant Nos. 11731003, 12031019, 12090012). Zeng Lian is partially supported by NNSF of China (Grant Nos. 11725105, 12090012)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, W., Lian, Z. Horseshoes for Anosov Systems on Fibers Driven by an Equicontinuous System. Acta. Math. Sin.-English Ser. 38, 281–290 (2022). https://doi.org/10.1007/s10114-022-0493-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-0493-z

Keywords

MR(2010) Subject Classification

Search

Navigation