Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-31T14:44:31.768Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

REFERENCES

[AY]Arnoux, P. & Yoccoz, J. C.. Construction de diffeomorphismes pseudo-Anosov. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 75–78.Google Scholar
[BO]Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978), 130–132.CrossRefGoogle Scholar
[DO]Douady, A. & Oesterlé, J.. Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris Sér. I Math. 290 (1980), 1135–1138.Google Scholar
[Fra1]Franks, J.. Anosov diffeomorphisms. Proc. Sympos. Pure Math. 14, 61–93.CrossRefGoogle Scholar
[Fra2]Franks, J.. Invariant sets of hyperbolic toral automorphisms. Amer. J. Math. 99 (1977), 1089–1095.CrossRefGoogle Scholar
[Fri]Fried, D.. Growth rate of surface homeomorphisms and flow equivalence. Ergod. Th. Dynam. Sys. 5 (1985), 539–563.CrossRefGoogle Scholar
[Ha]Hancock, S.. Construction of invariant sets for Anosov diffeomorphisms. J. London Math. Soc. 18 (1978), 339–348.Google Scholar
[Hi]Hirsch, M.. On invariant subsets of hyperbolic sets. In Essays in topology and related topics, 1970, pp. 126–146.CrossRefGoogle Scholar
[Ir1]Irwin, M.. The orbit of a Hölder continuous path under a hyperbolic toral automorphism. Ergod. Th. Dynam. Sys. 3 (1983), 345–349.CrossRefGoogle Scholar
[Ir2]Irwin, M.. Hölder continuous paths and hyperbolic toral automorphisms. Ergod. Th. Dynam. Sys. 6 (1986), 241–247.CrossRefGoogle Scholar
[LY]Ledrappier, F. & Young, L. S.. The metric entropy of diffeomorphisms part II: Relations between entropy, exponents and dimension. Ann. Math. 122 (1985), 540–574.CrossRefGoogle Scholar
[Le1]Levitt, G.. Propriètés homologiques des feuilletages des surfaces. C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 597–600.Google Scholar
[Le2]Levitt, G.. Feuilletages des surfaces. Thèse d'État, Université Paris 7, 1983.Google Scholar
[Ma1]Mañé, R.. Orbits of paths under hyperbolic toral automorphisms. Proc. Amer. Math. Soc. 73 (1979), 121–125.CrossRefGoogle Scholar
[Ma2]Mañé, R.. Invariant sets of Anosov diffeomorphisms. Invent. Math. 46 (1978), 147–152.CrossRefGoogle Scholar
[Pr]Przytycki, F.. Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors. Studia Math. 58 (1980), 199–213.CrossRefGoogle Scholar
[S]Shub, M.. Alexander cocycles and dynamics. Astérisque 81 (1977), 395–414.Google Scholar
[Ur]Urbański, M.. On the capacity of a continuum with a non dense orbit under a hyperbolic toral map, Studia Math. 81 (1985), 37–51.CrossRefGoogle Scholar