ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
Процессы Управления

On Absolute Nonshadowability of Transitive Maps


Sergey Tikhomirov

Saint-Petersburg State University 14th lane 29B, Vasilievsky Island,
St. Petersburg, 199178, Russia.
Max Planck Institute for Mathematics in the Science Inselstrasse 22,
04103 Leipzig, Germany.
Doctor of Science


We study shadowing property for random infinite pseudotrajectories of a continuous map f of a compact metric space. For the cases of transitive maps and transitive attractors we prove a dichotomy: either f satisfies shadowing property or random pseudotrajectory is shadowable with probability 0.

Ключевые слова


  1. Abdenur F. Attractors of generic diffeomorphisms are persistent. Nonlinearity 16 (2003), no. 1, 301-311
  2. Abdenur F., Diaz L. J. Pseudo-orbit shadowing in the $C^1$ topology. Discrete Contin. Dyn. Syst., 7 (2003), 223-245
  3. Anosov D. V. On a class of invariant sets of smooth dynamical systems. Proc. 5th Int. Conf. on Nonlin. Oscill. 2 (1970), 39-45
  4. Bowen R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes Math., vol. 470, Springer, Berlin, 1975
  5. Hammel S. M., Yorke J. A., Grebogi C. Do numerical orbits of chaotic dynamical processes represent true orbits. J. of Complexity 3 (1987), 136-145
  6. Hammel S. M., Yorke J. A., Grebogi C. Numerical orbits of chaotic processes represent true orbits. Bulletin of the American Mathematical Society 19 (1988), 465-469
  7. Hasselblatt B., Katok A. Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995
  8. Hasselblatt B., Katok A. A first course in dynamics. With a panorama of recent developments. Cambridge University Press, New York, 2003
  9. Leonov G. A. Strange Attractors and Classical Stability Theory. St. Petersburg University Press, 2009
  10. Palmer K. Shadowing in Dynamical Systems. Theory and Applications. Kluwer, Dordrecht, 2000
  11. Pilyugin S. Yu. Shadowing in Dynamical Systems. Lecture Notes Math., vol. 1706, Springer, Berlin, 1999
  12. Pilyugin S. Yu. Spaces of dynamical systems. De Gruyter Studies in Mathematical Physics, 3. De Gruyter, Berlin, 2012
  13. Pilyugin S. Yu. Theory of pseudo-orbit shadowing in dynamical systems. Diff. Eqs. 47 (2011), 1929-1938
  14. Pilyugin S. Yu., Rodionova A. A., Sakai K. Orbital and weak shadowing properties. Discrete Contin. Dyn. Syst., 9 (2003), 287-308
  15. Pilyugin S. Yu., Tikhomirov S. B. Lipschitz Shadowing implies structural stability. Nonlinearity 23 (2010), 2509-2515
  16. Sakai K. Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds. Osaka J. Math., 31 (1994), 373-386
  17. Tikhomirov S. Holder Shadowing on finite intervals. Ergodic Theory Dynam. Systems, 35 (2015), no. 6, 2000-2016
  18. Tikhomirov S. Shadowing in linear skew products. J. Math. Sci. (N. Y. ), 209 (2015), no. 6, 979-987
  19. Yuan G., Yorke J. An open set of maps for which every point is absolutely nonshadowable. Proc. Amer. Math. Soc. 128 (2000), 909-918

Полный текст (pdf)