Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T17:26:03.238Z Has data issue: false hasContentIssue false
  • English
  • Français

The Lyapunov dimension of a nowhere differentiable attracting torus

Published online by Cambridge University Press:  19 September 2008

James L. Kaplan ,
John Mallet-Paret  and
James A. Yorke
James L. Kaplan
Affiliation:
Department of Mathematics, Boston University, Boston, Massachusetts 02215, USA
John Mallet-Paret
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 and Lefschetz Centre for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA
James A. Yorke
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The fractal dimension of an attracting torus Tk in × Tk is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 2 , June 1984 , pp. 261 - 281
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Alexander, J. C. & Yorke, J. A.. Fat baker's transformations. Ergod. Th. & Dynam. Sys. 4, (1984), 1–24.Google Scholar
[2]Berry, M. V. & Lewis, Z. V.. Proc. Royal Soc. London A370 (1980), 459–484.Google Scholar
[3]Besicovitch, A. S. & Ursell, H. D.. J. London Math. Soc. 12 (1937) 18–25.CrossRefGoogle Scholar
[4]Corduneanu, C.. Almost Periodic Functions. Interscience: New York, 1968.Google Scholar
[5]Falconer, K. J.. The Geometry of Fractal Sets. Cambridge Univ. Press, in press.CrossRefGoogle Scholar
[6]Farmer, J. D., Ott, E. & Yorke, J. A.. The dimension of chaotic attractors. Phys. D. To appear.Google Scholar
[7]Frederickson, P., Kaplan, J. L., Yorke, E. D. & Yorke, J. A.. The Lyapunov dimension of strange attractors. J. Differential Equations. To appear.Google Scholar
[8]Hardy, G. H.. Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17 (1916), 301–325.Google Scholar
[9]Kaplan, J. L. & Yorke, J. A.. Chaotic behavior of multidimensional difference equations. In Functional Differential Equations and Approximation of Fixed Points (Peitgen, H. O. and Walther, H. O., eds.). Springer Verlag Lecture Notes in Math #730 (1979), 228–237.Google Scholar
[10]Kline, S. A.. J. London Math. Soc. 20 (1945) 79–86.Google Scholar
[11]Ledrappier, F.. Some relations between dimension and Lyapunov exponents. Commun. Math. Phys. 81 (1981), 229–238.Google Scholar
[12]Love, E. R. & Young, L. C.. Fundamental Math. 28 (1937) 243–257.Google Scholar
[13]Moser, J.. On a theorem of Anosov. J. Differential Equations 5 (1969), 411–490.Google Scholar