[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.
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.
[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.
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.