Communications in Mathematical Physics

, Volume 320, Issue 1, pp 101–119 | Cite as

Dimensions of Attractors in Pinched Skew Products

Article

Abstract

We study dimensions of strange non-chaotic attractors and their associated physical measures in so-called pinched skew products, introduced by Grebogi and his coworkers in 1984. Our main results are that the Hausdorff dimension, the pointwise dimension and the information dimension are all equal to one, although the box-counting dimension is known to be two. The assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable. Our findings confirm a conjecture by Ding, Grebogi and Ott from 1989.

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References

  1. 1.
    Grebogi C., Ott E., Pelikan S., Yorke J.A.: Strange attractors that are not chaotic. Physica D 13, 261–268 (1984)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Glendinning P.: Global attractors of pinched skew products. Dyn. Syst. 17, 287–294 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Keller G.: A note on strange nonchaotic attractors. Fund. Math. 151, 139–148 (1996)MathSciNetMATHGoogle Scholar
  4. 4.
    Prasad A., Negi S.S., Ramaswamy R.: Strange nonchaotic attractors. Int. J. Bif. Chaos 11(2), 291–309 (2001)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Haro A., Puig J.: Strange non-chaotic attractors in Harper maps. Chaos 16, 033127 (2006)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Jäger T.: The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Mem. Am. Math. Soc. 945, 1–106 (2009)MATHGoogle Scholar
  7. 7.
    Ding M., Grebogi C., Ott E.: Dimensions of strange nonchaotic attractors. Phys. Lett. A 137(4-5), 167–172 (1989)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Jäger T.: On the structure of strange nonchaotic attractors in pinched skew products. Erg. Th. Dyn. Syst. 27(2), 493–510 (2007)MATHCrossRefGoogle Scholar
  9. 9.
    Bjerklöv K.: Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations. Erg. Th. Dyn. Syst. 25, 1015–1045 (2005)MATHCrossRefGoogle Scholar
  10. 10.
    Bjerklöv K.: Dynamics of the quasiperiodic Schrödinger cocycle at the lowest energy in the spectrum. Commun. Math. Phys. 272, 397–442 (2005)ADSCrossRefGoogle Scholar
  11. 11.
    Bjerklöv K.: SNA’s in the quasi-periodic quadratic family. Commun. Math. Phys. 286(1), 137–161 (2009)ADSMATHCrossRefGoogle Scholar
  12. 12.
    Bjerklöv K.: Quasi-periodic perturbation of unimodal maps exhibiting an attracting 3-cycle. Nonlinearity 25, 683 (2012)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Alseda L., Misiurewicz M.: Attractors for unimodal quasiperiodically forced maps. J. Diff. Eqs. Appl. 14(10-11), 1175–1196 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jäger T.: Strange non-chaotic attractors in quasiperiodically forced circle maps. Commun. Math. Phys. 289(1), 253–289 (2009)ADSMATHCrossRefGoogle Scholar
  15. 15.
    Jäger T.: Quasiperiodically forced interval maps with negative Schwarzian derivative. Nonlinearity 16(4), 1239–1255 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Pesin, Ya.B.: Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 1997Google Scholar
  17. 17.
    Howroyd J.D.: On Hausdorff and packing dimension of product spaces. Math. Proc. Cambr. Phil. Soc. 119(4), 715–727 (1996)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Cutler C.D.: Some results on the behaviour and estimation of fractal dimensions of distributions on attractors. J. Stat. Phys. 62(3-4), 651–708 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Young L.S.: Dimension, entropy and Lyapunov exponents. Erg. Th. Dyn. Syst. 2(1), 109–124 (1982)MATHCrossRefGoogle Scholar
  20. 20.
    Zindulka, O.: Hentschel-Procaccia spectra in separable metric spaces. In: Real Anal. Exchange, 26th Summer Symposium Conference, suppl.:115–119, 2002, report on the Summer Symposium in Real Analysis XXVI, and unpublished note on http://mat.fsv.cvut.cz/zin/dulka/
  21. 21.
    Ledrappier F., Young L.-S.: The Metric Entropy of Diffeomorphisms: Part I: Characterization of Measures Satisfying Pesin’s Entropy Formula. Ann. Math. 122(3), 509–539 (1985)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Ledrappier F., Young L.-S.: The Metric Entropy of Diffeomorphisms: Part II: Relations between Entropy, Exponents and Dimension. Ann. Math. 122(3), 540–574 (1985)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ambrosio L., Kirchheim B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318(3), 527–555 (2000)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Boca Raton, FL: CRC-Press, 1992Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversität BremenBremenGermany
  2. 2.Department of MathematicsTU DresdenDresdenGermany

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