Topological Average Shadowing Property on Uniform Spaces

• Pirfalak, Farzaneh ^[1] ; Ahmadi, Seyyed Alireza ^[1] ; Kouhestani, Nader ^[2] ; Wu, Xinxing ^[1]
  1. [1] University of Sistan and Baluchestan

     University of Sistan and Baluchestan

     Irán

     
  2. [2] Southwest Petroleum University

     Southwest Petroleum University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00466-w
• Enlaces
  • Texto completo
• Resumen

  • We introduce topological definition of average shadowing property. We prove that topological average shadowing property implies topological chain transitivity. In particular it is proved that for a dynamical system with dense minimal points, the topological average shadowing property implies topological strong ergodicity.

• Referencias bibliográficas
  • Smaoui, N., Kanso, A.: Cryptography with chaos and shadowing. Chaos Solitons Fractals 42(4), 2312–2321 (2009). https://doi.org/10.1016/j.chaos.2009.03.128
  • Shao, H., Shi, Y., Zhu, H.: On distributional chaos in non-autonomous discrete systems. Chaos Solitons Fractals 107, 234–243 (2018). https://doi.org/10.1016/j.chaos.2018.01.005
  • Li, N., Wang, L.: Sensitivity and chaoticity on nonautonomous dynamical systems. Internet J. Bifur. Chaos Appl. Sci. Engrg. 30(10), 2050146...
  • Liu, G., Lu, T., Yang, X., Waseem, A.: Further discussion about transitivity and mixing of continuous maps on compact metric spaces. J. Math....
  • Li, R., Lu, T., Chen, G., Liu, G.: Some stronger forms of topological transitivity and sensitivity for a sequence of uniformly convergent...
  • Das, T., Lee, K., Richeson, D., Wiseman, J.: Spectral decomposition for topologically anosov homeomorphisms on noncompact and non-metrizable...
  • Shah, S., Das, T., Das, R.: Distributional chaos on uniform spaces, Qual. Theory Dyn. Syst. 19(1) (2020) Paper No. 4, 13. https://doi.org/10.1007/s12346-020-00344-x
  • Ahmadi, S.A.: Shadowing, ergodic shadowing and uniform spaces. Filomat 31, 5117–5124 (2017). https://doi.org/10.2298/FIL1716117A
  • Wu, X., Luo, Y., Ma, X., Lu, T.: Rigidity and sensitivity on uniform spaces. Topol. Its Appl. 252, 145–157 (2019). https://doi.org/10.1016/j.topol.2018.11.014
  • Wu, X., Liang, S., Ma, X., Lu, T., Ahmadi, S.A.: The mean sensitivity and mean equicontinuity in uniform spaces. Int. J. Bifur. Chaos (2020)....
  • Ahmadi, S.A., Wu, X., Feng, Z., Ma, X., Lu, T.: On the entropy points and shadowing in uniform spaces. Int. J. Bifur Chaos 28, 1850155 (2018)....
  • Good, C., Macías, S.: What is topological about topological dynamics? Discrete Contin. Dyn. Syst. 38, 1007–1031 (2018). https://doi.org/10.3934/dcds.2018043
  • Ahmadi, S.A.: On the topology of the chain recurrent set of a dynamical system. Appl. Gen. Topol. 15(2), 167–174 (2014). https://doi.org/10.4995/agt.2014.3050
  • Akin, E.: The General Topology of Dynamical Systems. Graduate Studies in Mathematics, vol. 1. American Mathematical Society, Providence (1993)
  • Akin, E., Wiseman, J.: Chain recurrence and strong chain recurrence on uniform spaces. In: Dynamical Systems and Random Processes, Vol. 736...
  • Morales, C.A., Sirvent, V.: Expansivity for measures on uniform spaces. Trans. Am. Math. Soc. 368(8), 5399–5414 (2016). https://doi.org/10.1090/tran/6555
  • Wu, X., Ma, X., Zhu, Z., Lu, T.: Topological ergodic shadowing and chaos on uniform spaces. Int. J. Bifur. Chaos 28(3), 1850043 (2018). https://doi.org/10.1142/S0218127418500438
  • Blank, M.L.: Metric properties of-trajectories of dynamical systems with stochastic behaviour. Ergodic Theory Dyn. Syst. 8(3), 365378 (1988)....
  • Dong, Y., Tian, X., Yuan, X.: Ergodic properties of systems with asymptotic average shadowing property. J. Math. Anal. Appl. 432(1), 53–73...
  • Gu, R.: The asymptotic average shadowing property and transitivity. Nonlinear Anal. Theory Methods Appl. 67(6), 1680–1689 (2007). https://doi.org/10.1016/j.na.2006.07.040
  • Niu, Y.: The average-shadowing property and strong ergodicity. J. Math. Anal. Appl. 376(2), 528–534 (2011). https://doi.org/10.1016/j.jmaa.2010.11.024
  • Niu, Y., Su, S.: On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property. Chaos Solitons Fractals 44(6),...
  • Niu, Y., Wang, Y., Su, S.: The asymptotic average shadowing property and strong ergodicity. Chaos Solitons Fractals 53, 34–38 (2013). https://doi.org/10.1016/j.chaos.2013.04.009
  • Wu, X., Wang, L., Liang, J.: The chain properties and average shadowing property of iterated function systems. Qual. Theory Dyn. Syst. 17,...
  • Ahmadi, S.A., Wu, X., Chen, G.: Topological chain and shadowing properties of dynamical systems on uniform spaces. Topol. Appl. 275, 107153...
  • Das, P., Das, T.: Various types of shadowing and specification on uniform spaces. J. Dyn. Control Syst. 24, 253–267 (2018). https://doi.org/10.1007/s10883-017-9388-1
  • Hart, K.P., Nagata, J.-I., Vaughan, J.E.: Encyclopedia of General Topology. Elsevier Science Publishers, Amsterdam (2004)
  • Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics. Springer, New York (2000)