Skip to main content
SpringerLink
Log in

Entropy and recurrent dimensions of discrete dynamical systems given by p-adic expansions

  • Research Articles
  • Published:
P-Adic Numbers, Ultrametric Analysis, and Applications Aims and scope Submit manuscript

Abstract

In this paper we study symbolic dynamical systems given by the shift mapping on the coefficient sequences of expansions of p-adic numbers. We associate the upper and lower recurrent dimensions with the topological entropies of these discrete dynamical systems by giving some inequalities representing the relationships among these parameters. Using these inequality relations, we estimate the topological entropies or recurrent dimensions of the various coefficient sequences, which have recurrent properties. For the case of Sturmian sequences we can estimate the positive gap values between upper and lower recurrent dimensions, which indicate the unpredictability of the orbits, if the irrational frequencies of the sequences are Liouville numbers.

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. B. Adamczewski and Y. Bugeaud, “On the complexity of algebraic numbers I. Expansions in integer bases,” Annals Math. 165, 547–565 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Morse and G. A. Hedlund, “Symbolic dynamics II-Sturmian trajectories,” Amer. J. Math. 62, 1–42 (1940).

    Article  MathSciNet  Google Scholar 

  3. K. Naito, “Dimension estimate of almost periodic attractors by simultaneous Diophantine approximation,” J. Diff. Equations 141, 179–200 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Naito, “Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations,” Trans. Amer. Math. Soc. 354(3), 1137–1151 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Naito, “Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions,” Discrete Cont. Dyn. Syst. 11, 449–488 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  6. K. Naito, “Classifications of irrational numbers and recurrent dimensions of quasi-periodic orbits,” J. Nonl. Anal. Convex Anal. 5, 169–185 (2004).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Graduate School of Science and Technology Kumamoto University, Kurokami 2-39-1, Kumamoto, Japan

    H. Inoue & K. Naito

Authors
  1. H. Inoue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Naito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Inoue.

Additional information

The text was submitted by the authors in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Inoue, H., Naito, K. Entropy and recurrent dimensions of discrete dynamical systems given by p-adic expansions. P-Adic Num Ultrametr Anal Appl 7, 157–167 (2015). https://doi.org/10.1134/S2070046615020077

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046615020077

Key words

Search

Navigation