Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T17:29:54.192Z Has data issue: false hasContentIssue false
  • English
  • Français

Hausdorff dimension for horseshoes

Published online by Cambridge University Press:  19 September 2008

Heather McCluskey and
Anthony Manning
Heather McCluskey
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
Anthony Manning
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
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.

We shall measure how thick a basic set of a C1 axiom A diffeomorphism of a surface is by the Hausdorff dimension of its intersection with an unstable manifold. This depends continuously on the diffeomorphism. Generically a C2 diffeomorphism has attractors whose Hausdorff dimension is not approximated by the dimension of its ergodic measures.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 2 , June 1983, pp. 251 - 260
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Besicovitch, A. S. & Moran, P. A. P.. The measure of product and cylinder sets. J. Lond. Math. Soc. 20 (1945), 110–120.Google Scholar
[2]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics No. 470. Springer Verlag: New York, 1975.CrossRefGoogle Scholar
[3]Bowen, R.. A horseshoe with positive measure. Inv. Math. 29 (1975), 203–204.CrossRefGoogle Scholar
[4]Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. IHES. 50 (1979), 11–25.CrossRefGoogle Scholar
[5]Bowen, R. & Ruelle, D.. The ergodic theory of axiom A flows. Inv. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[6]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic theory on compact spaces. Lecture Notes in Mathematics No. 527. Springer Verlag: Berlin, 1976.CrossRefGoogle Scholar
[7]Frederickson, P., Kaplan, J. & Yorke, J.. The dimension of the strange attractor for a class of difference systems. Preprint (1980).Google Scholar
[8]Gurevič, B. M. & Oseledeč, V. I.. Gibbs distributions and dissipativeness of U-diffeomorphisms. Sov. Math. Dokl. 14 (1973), 570–573.Google Scholar
[9]Hirsch, M. & Pugh, C.. Stable manifolds and hyperbolic sets. In Global Analysis. Proc. Symp. Pure Math. 14 (1970), 133–163.CrossRefGoogle Scholar
[10]Hurewicz, W. & Wallman, H.. Dimension Theory. Princeton University Press: Princeton, 1941.Google Scholar
[11]Manning, A.. A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 451–459.CrossRefGoogle Scholar
[12]Nitecki, Z.. Differentiable Dynamics. MIT Press: 1971.Google Scholar
[13]Pesin, Ya. B.. A formula for the Hausdorff dimension of a two-dimensional hyperbolic attractor. Funk. Anal. i ego Prilozh. 16 (1982), no. 4, 82–83.Google Scholar
[14]Ruelle, D.. Thermodynamic formalism. Addison-Wesley: Reading, 1978.Google Scholar
[15]Sinai, Ya. G.. The construction of Markov partitions. Func. Anal. Appl. 2 (1968), 70–80.CrossRefGoogle Scholar
[16]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[17]Smale, S.. The Ω-stability theorem. In Global Analysis, Proc. Symp. Pure Math. 14 (1970), 289–297.CrossRefGoogle Scholar
[18]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1976), 937–971.CrossRefGoogle Scholar
[19]Williams, R. F.. The DA maps of Smale and structural stability. In Global Analysis, Proc. Symp. Pure Math. 14 (1970), 329–334.CrossRefGoogle Scholar
[20]Young, L. S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Syst. 2 (1982), 109–124.CrossRefGoogle Scholar