Shadowing Property and Invariant Measures Having Full Supports

• Moriyasu Kazumine ^[3] ; Sakai Kazuhiro ^[1] ; Sumi Naoya ^[2]
  1. [1] Utsunomiya University

     Utsunomiya University

     Japón

     
  2. [2] Kumamoto University

     Kumamoto University

     Chuo-ku, Japón

     
  3. [3] Tokushima University

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00338-9
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• Resumen

  • We consider the shadowing property from the measure theoretical view point. Let f be a homeomorphism of a compact metric space. Then f has the shadowing property on the chain recurrent set if and only if there exists a shadowable f-invariant measure having full support. Suppose that f does not have the shadowing property. Then, if there exists a f-invariant measure having full support, then, generically, every f-invariant measure is not shadowable.

• Referencias bibliográficas
  • 1. Abdenur, F., Bonatti, C., Crovisier, S.: Nonuniform hyperbolicity for C1-generic diffeomorphisms. Isr. J. Math. 183, 1–60 (2011)
  • 2. Abdenur, F., Díaz, L.J.: Pseudo-orbit shadowing in the C1 topology. Discrete Contin. Dyn. Syst. 17, 223–245 (2007)
  • 3. Aoki, N.: Homeomorphisms with pseudo-orbit tracing property. Tokyo J. Math. 6, 329–334 (1983)
  • 4. Aoki, N.: Topics in Geral Topology. North-Holland Mathematical Library, vol. 41. North-Holland Publishing Co., Amsterdam (1989)
  • 5. Aoki, N., Hiraide, K.: Topological Theory of Dynamical Systems, Recent Advances. North-Holland Mathematical Library, vol. 52. North-Holland...
  • 6. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol. 527. Springer, Berlin...
  • 7. Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. 49,...
  • 8. Morales, C.A., Sirvent, V.F.: Expansive Measures. IMPA Mathematical Publications, vol. 29. Colóquio Brasileiro de Matemática, Instituto...
  • 9. Moriyasu, K., Sakai, K., Sumi, N.: Diffeomorphisms with shadowable measures. Axioms 7(4), 93 (2018). https://doi.org/10.3390/axioms7040093
  • 10. Pilyugin, S.Y., Sakai, K.: Shadowing and Hyperbolicity. Lecture Notes in Mathematics, vol. 2193. Springer, Berlin (2017)
  • 11. Robinson, C.: Stability theorems and hyperbolicity in dynamical systems. Rocky Mt. J. Math. 7, 425– 437 (1977)
  • 12. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. Studies in Advanced Mathematics, 2nd edn. CRC Press, Boca Raton...