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  • Ergodic Theory and Dynamical Systems, Volume 8, Issue 4
  • December 1988, pp. 523-529

On orbits of endomorphisms of tori and the Schmidt game

  • S. G. Dani (a1)
  • DOI:
  • Published online: 01 September 2008

We show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any xF and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.

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[1]S. G. Dani . Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv. 61 (1986), 636660.

[3]R. Mane . Orbits of paths under hyperbolic toral automorphisms. Proc. Amer. Math. Soc. 73 (1979), 121125.

[4]W. M. Schmidt . On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178199.

[5]W. M. Schmidt . Diophantine Approximation, Springer-Verlag: Berlin-Heidelberg-New York, 1980.

[6]P. Walters . An Introduction to Ergodic Theory, Springer-Verlag: Berlin-Heidelberg-New York, 1982.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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