Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T09:35:42.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Capacity of attractors

Published online by Cambridge University Press:  19 September 2008

Lai-Sang Young
Lai-Sang Young*
Affiliation:
Mathematics Institute, University of Warwick, England
*
Address for correspondence: Lai-Sang Young, Department of Mathematics, Michigan State University, East Lansing, MI 48824, 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.

Let f be a diffeomorphism of a manifold and Λ be an f-invariant set supporting an ergodic Borel probability measure μ with certain properties. A lower bound on the capacity of Λ is given in terms of the μ-Lyapunov exponents. This applies in particular to Axiom A attractors and their Bowen-Ruelle measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 3 , September 1981 , pp. 381 - 388
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. no. 470. Springer: Berlin, 1975.CrossRefGoogle Scholar
[2]Douady, A. & Oesterlé, J.. Dimension de Hausdorff des attracteurs. Comptes Rendus des Séances de L'Academie des Sciences 24 (1980), 1135–1138.Google Scholar
[3]Frederickson, P., Kaplan, J. & Yorke, J.. The dimension of the strange attractor for a class of difference systems. Preprint.Google Scholar
[4]Kaplan, J. & Yorke, J.. Chaotic behavior of multidimensional difference equations. Lecture Notes in Math. no. 730, pp. 228–237. Springer: Berlin, 1979.Google Scholar
[5]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S. 51 (1980), 137–174.Google Scholar
[6]Ledrappier, F.. Some relations between dimension and Lyapunov exponents. Comm. Math. Phys. 81 (1981), 229–238.Google Scholar
[7]Ledrappier, F.. In preparation.Google Scholar
[8]Mallet-Paret, J.. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. Journal Diff. Eqtns. 22 (1976), 331–348Google Scholar
[9]Mañé, R.. On the dimension of the compact invariant sets of certain nonlinear maps. Warwick Symp. Proc. Lecture Notes in Math. Springer: Berlin (to appear).Google Scholar
[10]Manning, A.. Hausdorff dimension of horseshoes. In preparation.Google Scholar
[11]Pesin, Ja.. Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. of the USSR—Izvestija 10 (1976), 6, 1261–1305.Google Scholar
[12]Pesin, Ja.. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surveys 32 (1977), 4, 55–114.Google Scholar
[13]Pugh, C. & Shub, M.. Ergodic attractors. In preparation.Google Scholar
[14]Ruelle, D.. Ergodic theory of differentiable dynamical systems. Publ. Math. I.H.E.S. 50 (1979), 27–58.Google Scholar
[15]Takens, F.. Detecting strange attractors in turbulence. Warwick Symp. Proc. Lecture Notes in Math. Springer: Berlin (to appear).Google Scholar