Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AVERAGE SHADOWING PROPERTIES ON COMPACT METRIC SPACES

  • Park Jong-Jin (Department of Mathematics Chonbuk National University) ;
  • Zhang Yong (Department of Mathematics Suzhou University)
  • Published : 2006.04.01
Download PDF
⟨ Previous Next ⟩

Abstract

We prove that if a continuous surjective map f on a compact metric space X has the average shadowing property, then every point x is chain recurrent. We also show that if a homeomorphism f has more than two fixed points on $S^1$, then f does not satisfy the average shadowing property. Moreover, we construct a homeomorphism on a circle which satisfies the shadowing property but not the average shadowing property. This shows that the converse of the theorem 1.1 in [6] is not true.

Keywords

References

  1. R. Bowen, Equilibrium states and the ergodic theory of axiom A diffeomorphims, New York: Spring-Verlag, (1975), 68-87
  2. M. L. Blank, Metric properties of E-trajectory of dynamical systems with stochastic behavior, Ergodic Theory and Dynamical systems 8 (1988), 365-378
  3. M. L. Blank, Small perturbations of chaotic dynamical systems, Russian Math survey 44 (1989), 1-33
  4. O. Plamenevskaya, Pseudo orbit tracing property and limit shadowing property on a circle, Vestnik St. Peterbug Univ. Math. 30 (2000), 1-9
  5. K. Sakai, Diffeomorphisms with the average-shadowing property on two dimensional closed manifold, Rocky mountain J. Math. 3 (2000), 1-69
  6. Y. Zhang, On the average-shadowing property, Acta Scientiarum Naturalium Universitaties Pekinensis 37 (2001), 648-651

Cited by

  1. Diffeomorphisms with C 1-stably average shadowing vol.29, pp.1, 2013, https://doi.org/10.1007/s10114-012-1162-4
  2. Symplectic diffeomorphisms with limit shadowing vol.10, pp.02, 2017, https://doi.org/10.1142/S1793557117500681
  3. h–Average-Shadowing Property and Chaos vol.11, pp.1-2, 2013, https://doi.org/10.1080/1726037X.2013.823806
  4. On partial shadowing of complete pseudo-orbits vol.411, pp.1, 2014, https://doi.org/10.1016/j.jmaa.2013.08.062
  5. On partial shadowing of complete pseudo-orbits vol.404, pp.1, 2013, https://doi.org/10.1016/j.jmaa.2013.02.068
  6. Some dynamical properties for free semigroup actions 2017, https://doi.org/10.1142/S0219493718500326
  7. Parameterized IFS with the Asymptotic Average Shadowing Property vol.15, pp.2, 2016, https://doi.org/10.1007/s12346-015-0184-6
  8. Asymptotic average shadowing property on compact metric spaces vol.69, pp.9, 2008, https://doi.org/10.1016/j.na.2007.08.058
  9. A Type of the Shadowing Properties for Generic View Points vol.7, pp.1, 2018, https://doi.org/10.3390/axioms7010018
  10. The ergodic shadowing property and homoclinic classes vol.2014, pp.1, 2014, https://doi.org/10.1186/1029-242X-2014-90