• Published : 2008.11.30


Let M be a generalized homogeneous compact space, and let Z(M) denotes the space of homeomorphisms of M with the $C^0$ topology. In this paper, we show that if the interior of the set of weak stable homeomorphisms on M is not empty then for any open subset W of Z(M) containing only weak stable homeomorphisms the orbital shadowing property is generic in W.


  1. T. Choi, S. Kim, and K. Lee, Weak inverse shadowing and genericity, Bull. Korean Math. Soc. 43 (2006), no. 1, 43-52
  2. B. A. Coomes, H. Ko¸cak, and K. J. Palmer, Periodic shadowing, Chaotic numerics (Geelong, 1993), 115-130, Contemp. Math., 172, Amer. Math. Soc., Providence, RI, 1994
  3. R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl. 189 (1995), no. 2, 409-423
  4. P. E. Kloeden and J. Ombach, Hyperbolic homeomorphisms and bishadowing, Ann. Polon. Math. 65 (1997), no. 2, 171-177
  5. P. Koscielniak and M. Mazur, Chaos and the shadowing property, Topology Appl. 154 (2007), no. 13, 2553-2557
  6. K. Kuratowski, Topology. Vol. II., Academic Press, New York-London; Panstwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warszawa, 1968
  7. M. Mazur, Weak shadowing for discrete dynamical systems on nonsmooth manifolds, J. Math. Anal. Appl. 281 (2003), no. 2, 657-662
  8. M. Mazur, Tolerance stability conjecture revisited, Topology Appl. 131 (2003), no. 1, 33-38
  9. S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999), no. 3, 253-266
  10. F. Takens, On Zeeman's tolerance stability conjecture, Lecture Notes in Mathematics 197, Spriger-Verlag, pp. 209-219, 1971
  11. P.Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., 668, Springer, Berlin, pp. 231-244, 1978

Cited by

  1. Weak Strictly Persistence Homeomorphisms and Weak Inverse Shadowing Property and Genericity vol.49, pp.3, 2009,