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The orbit of a Hölder continuous path under a hyperbolic toral automorphism

Published online by Cambridge University Press:  19 September 2008

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

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Let f:T3→T3 be a hyperbolic toral automorphism lifting to a linear automorphism with real eigenvalues. We prove that there is a Hölder continuous path in T3 whose orbit-closure is 1-dimensional. This strengthens results of Hancock and Przytycki concerning continuous paths, and contrasts with results of Franks and Mañé concerning rectifiable paths.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 3 , September 1983 , pp. 345 - 349
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Franks, J.. Invariant sets of hyperbolic toral automorphisms. Amer. J. Math. 99 (1977), 1089–1095.Google Scholar
[2]Hancock, S. G.. Construction of invariant sets for Anosov diffeomorphisms. J. London Math. Soc. (2) 18 (1978), 339–348.Google Scholar
[3]Hancock, S. G.. Invariant sets of Anosov diffeomorphisms. Thesis, Warwick University, 1979.Google Scholar
[4]Hirsch, M. W.. On invariant subsets of hyperbolic sets. In Essays on Topology and Related Topics. Springer-Verlag, 1970.CrossRefGoogle Scholar
[5]Mañé, R.. Orbits of paths under hyperbolic toral automorphisms. Proc. Amer. Math. Soc. 73 (1979), 121–125.CrossRefGoogle Scholar
[6]Przytycki, F.. Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors. Studio Math. 68 (1980), 199–213.CrossRefGoogle Scholar