Skip to main content
SpringerLink
Log in

Number theoretical peculiarities in the dimension theory of dynamical systems

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

Abstract

We show that dimensional theoretical properties of dynamical systems can considerably change because of number theoretical peculiarities of some parameter values.

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. J. C. Alexander and J. A. Yorke,Fat Baker’s transformations, Ergodic Theory and Dynamical Systems4 (1984), 1–23.

    MATH  MathSciNet  Google Scholar 

  2. M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J. P. Schreiber,Pisot and Salem Numbers, Birkhäuser Verlag, Basel, 1992.

    MATH  Google Scholar 

  3. M. Denker, C. Grillenberger and K. Sigmund,Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics527, Springer-Verlag, Berlin, 1976.

    MATH  Google Scholar 

  4. P. Erdös,On a family of symmetric Bernoulli convolutions, American Journal of Mathematics61 (1939), 974–976.

    Article  MathSciNet  Google Scholar 

  5. K. Falconer,The Hausdorff dimension of self-affine fractals, Mathematical Proceedings of the Cambridge Philosophical Society103 (1988), 339–350.

    Article  MATH  MathSciNet  Google Scholar 

  6. K. Falconer,Fractal Geometry — Mathematical Foundations and Applications, Wiley, New York, 1990.

    MATH  Google Scholar 

  7. A. M. Garsia,Arithmetic properties of Bernoulli convolutions, Transactions of the American Mathematical Society162 (1962), 409–432.

    Article  MathSciNet  Google Scholar 

  8. A. M. Garsia,Entropy and singularity of infinite convolutions, Pacific Journal of Mathematics13 (1963), 1159–1169.

    MATH  MathSciNet  Google Scholar 

  9. D. Gatzouras and S. P. Lalley,Hausdorff and box dimensions of certain self-affine fractals, Indiana University Mathematics Journal41 (1992), 533–568.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. E. Hutchinson,Fractals and self-similarity, Indiana University Mathematics Journal30 (1981), 271–280.

    Article  MathSciNet  Google Scholar 

  11. A. Katok and B. Hasselblatt,Introduction to Modern Theory of Dynamical Systems, Cambridge University Press, 1995.

  12. S. P. Lalley,Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution, Journal of the London Mathematical Society (2)57 (1999), 629–654.

    Google Scholar 

  13. A. Manning and H. McCluskey,Hausdorff dimension for horseshoes, Ergodic Theory and Dynamical Systems3 (1983), 251–260.

    MathSciNet  Google Scholar 

  14. C. McMullen,The Hausdorff dimension of general Sierpinski carpets, Nagoya Mathematical Journal96 (1984), 1–9.

    MATH  MathSciNet  Google Scholar 

  15. J. Neunhäuserer,Properties of some overlapping self-similar and some self-affine measures, Schwerpunktprogramm der deutschen Forschungsgemeinschaft: DANSE, Preprint 35/99; Acta Mathematica Hungarica92 (1–2) (2001), 143–161.

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Peres, B. Solomyak and W. Schlag,Sixty years of Bernoulli convolutions, Fractals and Stochastic II, Progress in Probability46 (2000), 95–106.

    MathSciNet  Google Scholar 

  17. Ya. Pesin,Dimension Theory in Dynamical Systems — Contemporary Views and Applications, University of Chicago Press, 1997.

  18. M. Pollicott and H. Weiss,The dimension of self-affine limit sets in the plane, Journal of Statistical Physics77 (1994), 841–860.

    Article  MATH  MathSciNet  Google Scholar 

  19. F. Przytychi and M. Urbanski,On Hausdorff dimension of some fractal sets, Studia Mathematica54 (1989), 218–228.

    Google Scholar 

  20. R. Salem,A remarkable class of algebraic integers, proof of a conjecture by Vijayarghavan, Duke Mathematical Journal11 (1944), 103–108.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Schmeling,A dimension formula for endomorphisms — The Belykh family, Ergodic Theory and Dynamical Systems18 (1998), 1283–1309.

    Article  MATH  MathSciNet  Google Scholar 

  22. B. Solomyak,On the random series Σ±λ i (an Erdös problem), Annals of Mathematics142 (1995), 611–625.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Solomyak,Measures and dimensions for some fractal families, Proceedings of the Cambridge Philosophical Society124 (1998), 531–546.

    Article  MATH  MathSciNet  Google Scholar 

  24. B. Solomyak and K. Simon,Dimension of horseshoes in ℞ 3,Ergodic Theory and Dynamical Systems19 (1999), 1345–1363.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Technische Universität Clausthal, Arnold-Sommerfeld-Str., 38678, Clausthal-Zellerfeld, Germany

    J. Neunhäuserer

Authors
  1. J. Neunhäuserer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by “DFG-Schwerpunktprogramm — Dynamik: Analysis, effiziente Simulation und Ergodentheorie”.

We refer to the book of Falconer [6] for an introduction to dimension theory and recommend the book of Pesin [17] for the dimension theory of dynamical systems.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Neunhäuserer, J. Number theoretical peculiarities in the dimension theory of dynamical systems. Isr. J. Math. 128, 267–283 (2002). https://doi.org/10.1007/BF02785428

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02785428

Keywords

Search

Navigation