Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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DYNAMICAL SYSTEMS WITH SPECIFICATION

  • Lee, Keonhee (Department of Mathematics Chungnam National University) ;
  • Tajbakhsh, Khosro (Department of Mathematics Tarbiat Modares University)
  • Received : 2014.09.20
  • Accepted : 2014.10.15
  • Published : 2015.02.15
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Abstract

In this paper we prove that $C^1$-generically, if a diffeomorphism f on a closed $C^{\infty}$ manifold M satisfies weak specification on a locally maximal set ${\Lambda}{\subset}M$ then ${\Lambda}$ is hyperbolic for f. As a corollary we obtain that $C^1$-generically, every diffeomorphism with weak specification is Anosov.

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References

  1. F. Abdenur and L. Diaz, Pseudo-orbit shadowing in the $C^1$ topology, Disc. Contin. Dynam. Syst. 17 (2007), 223-245.
  2. N. Aoki, M. Dateyama, and M. Komuro, Solenoidal automorphisms and specification, Monatshefte fur Mathematik 93 (1982), 79-110. https://doi.org/10.1007/BF01301397
  3. R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397.
  4. S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. IHES 104 (2006), 87-141. https://doi.org/10.1007/s10240-006-0002-4
  5. M. Denker, C. Girllenberger, and K. Sigmund, Ergodic Theory On Compact spaces, Lecture Note in Math. 527 Springer Verlag, Berlin, 1976.
  6. M. Lampart and P. Oprocha, Shift spaces, ${\omega}$-chaos and specification property, Topology and its Appl. 156 (2009), 2979-2985. https://doi.org/10.1016/j.topol.2009.04.063
  7. D. A. Lind, Split skew products, a related functional equation, and specification, Israel J. of Math. 30 (1978), 236-254. https://doi.org/10.1007/BF02761073
  8. D. A. Lind, Ergodic Group Automorphisms and specification, Ergodic theory, Proc. Conf., Math. forschungsinst., Oberwolfach, 1978, 93-104,Lecture Note in Math. 729 Springer Verlag, Berlin, 1979.
  9. R. Mane, An ergodic closing lemma, Ann. Math. 116 (1982), 503-540. https://doi.org/10.2307/2007021
  10. D. Ruelle, Statical mecjhanics on a compact set with $Z^{\mu}$-action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 185 (1973), 273-251.
  11. K. Sakai, N. Sumi, and K. Yamamoto, Diffeomorphisms satisfying the specification property, Proc. Amer. Math. Soc. 138 (2010), 315-321. https://doi.org/10.1090/S0002-9939-09-10085-0
  12. K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974), 285-299. https://doi.org/10.1090/S0002-9947-1974-0352411-X
  13. K. Yamamoto, On the weaker forms of the specification property and their applications, Proc. Amer. Math. Soc. 137 (2009), 3807-3814. https://doi.org/10.1090/S0002-9939-09-09937-7