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Some compact invariant sets for hyperbolic linear automorphisms of torii

Published online by Cambridge University Press:  19 September 2008

Albert Fathi
Albert Fathi
Affiliation:
Department of Mathematics, Walker Hall, University of Florida, Gainesville, Florida 32611, USA
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Abstract

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If the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 2 , June 1988 , pp. 191 - 204
Copyright
Copyright © Cambridge University Press 1988

References

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