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Year 2021, Volume: 50 Issue: 5, 1371 - 1383, 15.10.2021

Abstract

References

  • [1] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16, 1421–1433, 2003.
  • [2] T. Arai and N. Chinen, P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy, Topology Appl. 154, 1254–1262, 2007.
  • [3] W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79, 81–92, 1975.
  • [4] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547, 51–68, 2002.
  • [5] M.L. Blank, Metric properties of ϵ-trajectories of dynamical systems with stochastic behaviour, Ergodic Theory Dynam. Systems, 8, 365–378, 1988.
  • [6] R. Das and M. Garg, Average chain transitivity and the almost average shadowing property, Commun. Korean Math. Soc. 32, 201–214, 2017.
  • [7] D.A. Dastjerdi and M. Hosseini, Sub-shadowings, Nonlinear Anal. 72, 3759–3766, 2010.
  • [8] Y. Dong, X. Tian and X. Yuan, Ergodic properties of systems with asymptotic average shadowing property, J. Math. Anal. Appl. 432, 53–73, 2015.
  • [9] A. Fakhari and F.H. Ghane, On shadowing: ordinary and ergodic, J. Math. Anal. Appl. 364, 151–155, 2010.
  • [10] M. Garg and R. Das, Relations of the almost average shadowing property with ergod- icity and proximality, Chaos Solitons Fractals, 91, 430–433, 2016.
  • [11] R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal. 67, 1680–1689, 2007.
  • [12] R. Gu and W. Guo, On mixing property in set-valued discrete systems, Chaos Solitons Fractals 28, 747–754, 2006.
  • [13] J.L.G. Guirao, D. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hy- perspaces, Nonlinear Anal. 71, 1–8, 2009.
  • [14] W. Huang and X. Ye, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl. 117, 259–272, 2002.
  • [15] M. Kulczycki, A unified approach to theories of shadowing, Regul. Chaotic Dyn. 19, 310–317, 2014.
  • [16] M. Kulczycki, D. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fund. Math. 224, 241–278, 2014.
  • [17] M. Kulczycki and P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property, Fund. Math. 212, 35–52, 2011.
  • [18] K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst. 13, 533–540, 2005.
  • [19] R. Li, A note on chaos via Furstenberg family couple, Nonlinear Anal. 72, 2290–2299, 2010.
  • [20] R. Li, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul. 17, 2815–2823, 2012.
  • [21] R. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45, 753–758, 2012.
  • [22] R. Li, A note on chaos and the shadowing property, Int. J. Gen. Syst. 45, 675–688, 2016.
  • [23] R. Li and X. Zhou, A note on ergodicity of systems with the asymptotic average shadowing property, Discrete Dyn. Nat. Soc. 2011, 6 pages, 2011.
  • [24] R. Li and X. Zhou, A note on chaos in product maps, Turkish J. Math. 37, 665–675, 2013.
  • [25] T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82, 985– 992, 1975.
  • [26] M. Miyazawa, Chaos and entropy for graph maps, Tokyo J. Math. 27, 221–225, 2004.
  • [27] S.B. Nadler, Jr., Continuum theory: An introduction, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1992.
  • [28] P. Oprocha, Families, filters and chaos, Bull. Lond. Math. Soc. 42, 713–725, 2010.
  • [29] P. Oprocha, D. Ahmadi Dastjerdi and M. Hosseini, On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl. 411, 454–463, 2014.
  • [30] A. Peris, Set-valued discrete chaos, Chaos Solitons Fractals, 26, 19–23, 2005.
  • [31] S.Y. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.
  • [32] K. Sakai, Various shadowing properties for positively expansive maps, Topology Appl. 131, 15–31, 2003.
  • [33] J. de Vries, Elements of topological dynamics, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993.
  • [34] X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29, 1942–1972, 2016.
  • [35] Y. Wu and X. Xue, Shadowing property for induced set-valued dynamical systems of some expansive maps, Dynam. Systems Appl. 19, 405–414, 2010.

Chaotic behaviour of maps possessing the almost average shadowing property

Year 2021, Volume: 50 Issue: 5, 1371 - 1383, 15.10.2021

Abstract

In this paper, we investigate the chaotic behaviour of maps having the almost average shadowing property by obtaining the relationship of the almost average shadowing property with different kinds of chaos. Moreover, we relate the notion of almost average shadowing property with some other types of shadowing properties, for instance, ergodic shadowing, $\mathcal{F}_{\underline{d}}$-shadowing and $\underline{d}$-shadowing. We also study the notion of almost average shadowing property for maps induced on hyperspaces. Our study is supported by providing counter-examples wherever necessary.

References

  • [1] E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16, 1421–1433, 2003.
  • [2] T. Arai and N. Chinen, P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy, Topology Appl. 154, 1254–1262, 2007.
  • [3] W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79, 81–92, 1975.
  • [4] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547, 51–68, 2002.
  • [5] M.L. Blank, Metric properties of ϵ-trajectories of dynamical systems with stochastic behaviour, Ergodic Theory Dynam. Systems, 8, 365–378, 1988.
  • [6] R. Das and M. Garg, Average chain transitivity and the almost average shadowing property, Commun. Korean Math. Soc. 32, 201–214, 2017.
  • [7] D.A. Dastjerdi and M. Hosseini, Sub-shadowings, Nonlinear Anal. 72, 3759–3766, 2010.
  • [8] Y. Dong, X. Tian and X. Yuan, Ergodic properties of systems with asymptotic average shadowing property, J. Math. Anal. Appl. 432, 53–73, 2015.
  • [9] A. Fakhari and F.H. Ghane, On shadowing: ordinary and ergodic, J. Math. Anal. Appl. 364, 151–155, 2010.
  • [10] M. Garg and R. Das, Relations of the almost average shadowing property with ergod- icity and proximality, Chaos Solitons Fractals, 91, 430–433, 2016.
  • [11] R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal. 67, 1680–1689, 2007.
  • [12] R. Gu and W. Guo, On mixing property in set-valued discrete systems, Chaos Solitons Fractals 28, 747–754, 2006.
  • [13] J.L.G. Guirao, D. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hy- perspaces, Nonlinear Anal. 71, 1–8, 2009.
  • [14] W. Huang and X. Ye, Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl. 117, 259–272, 2002.
  • [15] M. Kulczycki, A unified approach to theories of shadowing, Regul. Chaotic Dyn. 19, 310–317, 2014.
  • [16] M. Kulczycki, D. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fund. Math. 224, 241–278, 2014.
  • [17] M. Kulczycki and P. Oprocha, Properties of dynamical systems with the asymptotic average shadowing property, Fund. Math. 212, 35–52, 2011.
  • [18] K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst. 13, 533–540, 2005.
  • [19] R. Li, A note on chaos via Furstenberg family couple, Nonlinear Anal. 72, 2290–2299, 2010.
  • [20] R. Li, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul. 17, 2815–2823, 2012.
  • [21] R. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45, 753–758, 2012.
  • [22] R. Li, A note on chaos and the shadowing property, Int. J. Gen. Syst. 45, 675–688, 2016.
  • [23] R. Li and X. Zhou, A note on ergodicity of systems with the asymptotic average shadowing property, Discrete Dyn. Nat. Soc. 2011, 6 pages, 2011.
  • [24] R. Li and X. Zhou, A note on chaos in product maps, Turkish J. Math. 37, 665–675, 2013.
  • [25] T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82, 985– 992, 1975.
  • [26] M. Miyazawa, Chaos and entropy for graph maps, Tokyo J. Math. 27, 221–225, 2004.
  • [27] S.B. Nadler, Jr., Continuum theory: An introduction, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1992.
  • [28] P. Oprocha, Families, filters and chaos, Bull. Lond. Math. Soc. 42, 713–725, 2010.
  • [29] P. Oprocha, D. Ahmadi Dastjerdi and M. Hosseini, On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl. 411, 454–463, 2014.
  • [30] A. Peris, Set-valued discrete chaos, Chaos Solitons Fractals, 26, 19–23, 2005.
  • [31] S.Y. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.
  • [32] K. Sakai, Various shadowing properties for positively expansive maps, Topology Appl. 131, 15–31, 2003.
  • [33] J. de Vries, Elements of topological dynamics, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993.
  • [34] X. Wu, P. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29, 1942–1972, 2016.
  • [35] Y. Wu and X. Xue, Shadowing property for induced set-valued dynamical systems of some expansive maps, Dynam. Systems Appl. 19, 405–414, 2010.
There are 35 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mukta Garg This is me 0000-0001-5863-3634

Ruchi Das 0000-0002-1889-9988

Publication Date October 15, 2021
Published in Issue Year 2021 Volume: 50 Issue: 5

Cite

APA Garg, M., & Das, R. (2021). Chaotic behaviour of maps possessing the almost average shadowing property. Hacettepe Journal of Mathematics and Statistics, 50(5), 1371-1383.
AMA Garg M, Das R. Chaotic behaviour of maps possessing the almost average shadowing property. Hacettepe Journal of Mathematics and Statistics. October 2021;50(5):1371-1383.
Chicago Garg, Mukta, and Ruchi Das. “Chaotic Behaviour of Maps Possessing the Almost Average Shadowing Property”. Hacettepe Journal of Mathematics and Statistics 50, no. 5 (October 2021): 1371-83.
EndNote Garg M, Das R (October 1, 2021) Chaotic behaviour of maps possessing the almost average shadowing property. Hacettepe Journal of Mathematics and Statistics 50 5 1371–1383.
IEEE M. Garg and R. Das, “Chaotic behaviour of maps possessing the almost average shadowing property”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, pp. 1371–1383, 2021.
ISNAD Garg, Mukta - Das, Ruchi. “Chaotic Behaviour of Maps Possessing the Almost Average Shadowing Property”. Hacettepe Journal of Mathematics and Statistics 50/5 (October 2021), 1371-1383.
JAMA Garg M, Das R. Chaotic behaviour of maps possessing the almost average shadowing property. Hacettepe Journal of Mathematics and Statistics. 2021;50:1371–1383.
MLA Garg, Mukta and Ruchi Das. “Chaotic Behaviour of Maps Possessing the Almost Average Shadowing Property”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, 2021, pp. 1371-83.
Vancouver Garg M, Das R. Chaotic behaviour of maps possessing the almost average shadowing property. Hacettepe Journal of Mathematics and Statistics. 2021;50(5):1371-83.