Hostname: page-component-6766d58669-h8lrw Total loading time: 0 Render date: 2026-05-22T17:56:55.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Hölder continuous paths and hyperbolic toral automorphisms

Published online by Cambridge University Press:  19 September 2008

M. C. Irwin
M. C. Irwin
Affiliation:
Department of Pure Mathematics, Liverpool University, Liverpool, L69 3BX, 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.

Let f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 2 , June 1986 , pp. 241 - 257
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978), 130–132.CrossRefGoogle Scholar
[2]Franks, J.. Invariant sets of hyperbolic toral automorphisms. Amer. J. Math. 99 (1977), 1089–1095.CrossRefGoogle Scholar
[3]Hancock, S.G.. Construction of invariant sets for Anosov diffeomorphisms. J. London Math. Soc. (2) 18 (1978), 339–348.Google Scholar
[4]Hancock, S. G.. Invariant sets of Anosov diffeomorphisms. Thesis. Warwick University, 1979.Google Scholar
[5]Hirsch, M. W.. On invariant subsets of hyperbolic sets. In Essays on Topology and Related Topics. Springer-Verlag, 1970.Google Scholar
[6]Hurewicz, W. and Wallman, H.. Dimension Theory. Princeton University Press, 1941.Google Scholar
[7]Irwin, M. C.. The orbit of a Hölder continuous path under a hyperbolic toral automorphism. Ergod. Th. & Dynam. Sys. 3 (1983), 345–349.CrossRefGoogle Scholar
[8]Mañé, R.. Orbits of paths under hyperbolic toral automorphisms. Proc. Amer. Math Soc. 73 (1979), 121–125.CrossRefGoogle Scholar
[9]Przytycki, F.. Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors. Studia Math. 68 (1980), 199–213.CrossRefGoogle Scholar
[10]Urbanski, M.. On the capacity of a continuum with a non-dense orbit under a hyperbolic toral automorphism. Studia Math. 81 (1985), 37–51.CrossRefGoogle Scholar