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Characterizing ergodicity of induced hyperspace dynamical systems

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Abstract

Let E be Hausdorff locally compact second countable spaces (HLCSC) and \((2^{E}, 2^{f})\) (hit-or-miss topology equipped) be hyperspace dynamical system induced by a given dynamical system (E, f). In this paper, the concepts of topologically co-compact ergodicity (resp. topologically co-compact strong ergodicity) and topologically co-compact double ergodicity (resp. topologically co-compact double strong ergodicity) are introduced for dynamical systems. For any HLCSC system (E, f), these three conditions on (E, f) are, respectively, equivalent to topological ergodicity (resp. topologically strong ergodicity) and topological double ergodicity (resp. topological double strong ergodicity) on \((2^{E}, 2^{f})\). The concept of topologically co-compact exact (c-exact) is also introduced, and we show that if f is perfect and c-exact, then \(2^{f}:{\mathcal {F}}_{00}\rightarrow {\mathcal {F}}_{00}\) is topologically exact, where \({\mathcal {F}}_{00}=\{F\in {\mathcal {F}}_{0}:\)F is finite\(\}\) and \({\mathcal {F}}_{0}=2^{E}\). In addition, other noticeable properties of topologically co-compact ergodicity (resp. topologically co-compact strong ergodicity) and topologically co-compact double ergodicity (resp. topologically co-compact double strong ergodicity) are studied.

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References

  1. Akin, E.: The General Topology of Dynamical Systems. Graduate Studies in Mathematics, vol. 1. American Mathematical Society, Providence (1993)

  2. Banks, J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals 25, 681–685 (2005)

    Article  MathSciNet  Google Scholar 

  3. Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publisher, Dordrecht (1993)

    Book  Google Scholar 

  4. Beer, G.: On the fell topology. Set Valued Anal. 1, 69–80 (1993)

    Article  MathSciNet  Google Scholar 

  5. Bernardes Jr., N.C., Peris, A., Rodenas, F.: Set-valued chaos in linear dynamics. Integral Equ. Oper. Theory 88(4), 451–463 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bernardes Jr., N.C., Vermersch, R.: Hyperspace dynamics of generic maps of the Cantor space. Can. J. Math. 67(2), 330–349 (2015)

    Article  MathSciNet  Google Scholar 

  7. Engelking, R.: General Topology. PWN-Polish Scientific Publishers, Warszawa (1977) (or Heldermann, Berlin, 1989)

  8. Fedeli, A.: On chaotic set-valued discrete dynamical systems. Chaos Solitons Fractals 23, 1381–1384 (2005)

    Article  MathSciNet  Google Scholar 

  9. Fell, J.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Am. Math. Soc. 13, 472–476 (1962)

    Article  MathSciNet  Google Scholar 

  10. Fu, H., Xing, Z.: Mixing properties of set-valued maps on hyperspaces via Furstenberg families. Chaos Solitons Fractals 45(4), 439–443 (2012)

    Article  MathSciNet  Google Scholar 

  11. Gu, R., Guo, W.: On mixing property in set-valued discrete systems. Chaos Solitons Fractals 28, 747–754 (2006)

    Article  MathSciNet  Google Scholar 

  12. Gu, R.: The large deviations theorem and ergodicity. Chaos Solitons Fractals 34, 1387–1392 (2007)

    Article  MathSciNet  Google Scholar 

  13. Gu, R.: On ergodicity of systems with the asymptotic average shadowing property. Comput. Math. Appl. 55, 1137–1141 (2008)

    Article  MathSciNet  Google Scholar 

  14. Guirao, J.L.G., Kwietniak, D., Lampart, M., Oprocha, P., Peris, A.: Chaos on hyperspaces. Nonlinear Anal. 71(1–2), 1–8 (2009)

    Article  MathSciNet  Google Scholar 

  15. Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fractals 33, 76–86 (2007)

    Article  MathSciNet  Google Scholar 

  16. Li, J., Yan, K., Ye, X.: Recurrence properties and disjointness on the induced spaces. Discret. Contin. Dyn. Syst. 35(3), 1059–1073 (2015)

    Article  MathSciNet  Google Scholar 

  17. Li, R.: Topological ergodicity, transitivity and chaos of the set-valued maps. J. NanJing Univ. Math. Biquarterly 25, 114–121 (2008). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  18. Li, R., Shi, Y.: Several sufficient conditions for sensitive dependence on initial conditions. Nonlinear Anal. A 72, 2716–2720 (2010)

    Article  MathSciNet  Google Scholar 

  19. Li, R., Zhou, X.: A note on ergodicity of systems with the asymptotic average shadowing property. Discret. Dyn. Nat. Soc. 2011, 6 (2011) (article ID 360583)

  20. Li, R., Shi, Y.: Stronger forms of sensitivity for measure-preserving maps and semi-flows on probability spaces. Abstr. Appl. Anal. 2014, 10 (2014) (article ID 769523)

  21. Liao, G., Wang, L., Zhang, Y.: Transitivity, mixing and chaos for a class of set-valued mappings. Sci. China Ser. A 49(1), 1–8 (2006)

    Article  MathSciNet  Google Scholar 

  22. Liao, G., Ma, X., Wang, L.: Individual chaos implies collective chaos for weakly mixing discrete dynamical systems. Chaos Solitons Fractals 32, 604–608 (2007)

    Article  MathSciNet  Google Scholar 

  23. Ma, X., Hou, B., Liao, G.: Chaos in hyperspace system. Chaos Solitons Fractals 40, 653–660 (2009)

    Article  MathSciNet  Google Scholar 

  24. Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)

    MATH  Google Scholar 

  25. Nogura, T., Shakhmatov, D.: When does the Fell topology on a hyperspace of closed sets coincide with meet of the upper Kuratowski and the lower Vietoris topology? Topol. Appl. 70, 213–243 (1996)

    Article  Google Scholar 

  26. Niu, Y., Wang, Y.: The central limit theorem and ergodicity. Stat. Probab. Lett. 80, 1180–1184 (2010)

    Article  MathSciNet  Google Scholar 

  27. Peris, A.: Set-valued discrete chaos. Chaos Solitons Fractals 26, 19–23 (2005)

    Article  MathSciNet  Google Scholar 

  28. Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals 17, 99–104 (2003)

    Article  MathSciNet  Google Scholar 

  29. Román-Flores, H., Chalco-Cano, Y.: Robinson’s chaos in set-valued discrete systems. Chaos Solitons Fractals 25, 33–42 (2005)

    Article  MathSciNet  Google Scholar 

  30. Salzburg, W., Wien, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79, 81–92 (1975)

    Article  MathSciNet  Google Scholar 

  31. Stoyan, D.: Models and statistics. Int. Stat. Rev. 66, 1–27 (1998)

    Article  Google Scholar 

  32. Wang, H., Xiong, J.: Some remarks on topologically ergodic maps. Acta Math. Sin. 47, 859–866 (2004). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  33. Wang, Y., Wei, G.: Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. Topol. Appl. 155, 56–68 (2007)

    Article  MathSciNet  Google Scholar 

  34. Wang, Y., Wei, G., Campbell, W.H.: Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. Topol. Appl. 156, 803–811 (2009)

    Article  MathSciNet  Google Scholar 

  35. Wei, G.: Contributions to distributions of random sets on Polish spaces. Ph.D. thesis. New Mexico State University, Las Cruces, New Mexico (1999)

  36. Wei, G., Wang, Y.: On metrization of the hit-or-miss topology using Alexandroff compactification. In: Nguyen, H.T., Miranda, E. (eds.) Random Sets and Imprecise Probabilities. International Journal of Approximate Reasoning, vol. 46, no. 1, pp. 47–64 (2007)

  37. Wu, X., Oprocha, P., Chen, G.: On various definitions of shadowing with average error in tracing. Nonlinearity 29, 1942–1972 (2016)

    Article  MathSciNet  Google Scholar 

  38. Wu, X., Chen, G.: Central limit theorem and chaoticity. Stat. Probab. Lett. 92, 137–142 (2014)

    Article  MathSciNet  Google Scholar 

  39. Wu, X., Chen, G.: On the large deviations theorem and ergodicity. Commun. Nonlinear Sci. Numer. Simul. 30, 243–247 (2016)

    Article  MathSciNet  Google Scholar 

  40. Wu, X., Luo, Y., Ma, X., Lu, T.: Rigidity and sensitivity on uniform spaces. Topol. Appl. 252, 145–157 (2019)

    Article  MathSciNet  Google Scholar 

  41. Wu, X., Wang, X.: On the iteration properties of large deviations theorem. Int. J. Bifurc. Chaos 26, 1650054 (2016)

    Article  MathSciNet  Google Scholar 

  42. Wu, X., Wang, J., Chen, G.: \(\mathscr {F}\)-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429(1), 16–26 (2015)

    Article  MathSciNet  Google Scholar 

  43. Wu, X., Wang, X., Chen, G.: On the large deviations theorem of weaker types. Int. J. Bifurc. Chaos 27, 1750127 (2017)

    Article  MathSciNet  Google Scholar 

  44. Wu, X., Zhang, X., Ma, X.: Various shadowing in linear dynamical systems. Int. J. Bifurc. Chaos (accepted for publication)

  45. Yang, R.: Topologically ergodic maps. Acta Math. Sin. 44, 1063–1068 (2001). (in Chinese)

    Google Scholar 

  46. Yang, R.: Topological ergodicity and topological double ergodicity. Acta Math. Sin. 46, 555–560 (2003). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, G., Zeng, F., Liu, X.: Devaney’s chaotic on induced maps of hyperspace. Chaos Solitons Fractals 27, 471–475 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the referees very much for many valuable comments and suggestions which greatly improved the writing of the paper. This research was supported the Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province (2018RZJ03), the Opening Project of Bridge Non-destruction Detecting and Engineering Computing Key Laboratory of Sichuan Province (2018QZJ03) and the Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City (Grant 2010C3112005).

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  1. School of Mathematic and Computer Science, Guangdong Ocean University, Zhanjiang, 524025, People’s Republic of China

    Risong Li

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Li, R. Characterizing ergodicity of induced hyperspace dynamical systems. Bol. Soc. Mat. Mex. 26, 223–238 (2020). https://doi.org/10.1007/s40590-018-0229-3

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