Abstract
In this paper, we study variants of transitivity and sensitivity via Furstenberg families for iterated function systems (IFSs). Using the concept of skew product transformation of an IFS, we obtain results related to characterizations of the concepts studied. Also results regarding the inverse, conjugacy and product of IFSs are studied. Wherever necessary examples and counterexamples are provided related to the results obtained.
Similar content being viewed by others
References
Akin, E.: Recurrence in Topological Dynamical Systems: families and Ellis actions. Univ. Ser. Math. Plenum Press, New York (1997)
Bahabadi, A.Z.: On chaos for iterated function systems. Asian Eur. J. Math. 11(4), 1850054 (2018)
Barnsley, M. F.: Fractals Everywhere, second ed. Academic Press Professional, Boston, MA, (1993). Revised with the assistance of and with a foreword by Hawley Rising, III
Birkhoff, G.D.: Collected Mathematical Papers, vol. 2. Dover Publications, USA (1950)
Chen, Z., Li, J., Lü, J.: Point transitivity, \(\Delta \)-transitivity and multi-minimality. Ergodic Theory Dyn. Syst. 35(5), 1423–1442 (2015)
Cohen, H.A.: The application of IFS iterated function systems to image analysis. In: Proceedings of the VEE International Conference on Image Processing ICIP vol. 89, pp. 5–8 (1989)
Değirmenci, N., Koçak, S.: Chaos in product maps. Turkish J. Math. 34(4), 593–600 (2010)
Devaney, R.L.: An introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA (1989)
Elton, J.H., Piccioni, M.: Iterated function systems arising from recursive estimation problems. Probab. Theory Related Fields 91(1), 103–114 (1992)
Fatehi Nia, M.: Parameterized IFS with the asymptotic average shadowing property. Qual. Theory Dyn. Syst. 15(2), 367–381 (2016)
Forte, B., Vrscay, E.R.: Solving the inverse problem for function/image approximation using iterated function systems. II. Algorithm and computations. Fractals 2(3), 335–346 (1994)
Ghane, F.H., Rezaali, E., Sarizadeh, A.: Sensitivity of iterated function systems. J. Math. Anal. Appl. 469(2), 493–503 (2019)
Hui, H., Ma, D.: Some dynamical properties for free semigroup actions. Stoch. Dyn. 18, 4 (2018). 1850032, 20
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Li, J.: Transitive points via Furstenberg family. Topol. Appl. 158(16), 2221–2231 (2011)
Li, R., Zhao, Y., Wang, H.: Furstenberg families and chaos on uniform limit maps. J. Nonlinear Sci. Appl. 10(2), 805–816 (2017)
Ma, C., Zhu, P.: A remark on sensitivity and Li-Yorke sensitivity of iterated function systems. Qual. Theory Dyn. Syst. 18(1), 1–9 (2019)
Ma, C., Zhu, P., Li, R.: On iteration invariants for \(( F_1, F_2)\)-sensitivity and weak \(( F_1, F_2)\)-sensitivity of non-autonomous discrete systems. J. Nonlinear Sci. Appl. 9(11), 5772–5779 (2016)
Montrucchio, L., Privileggi, F.: Fractal steady states in stochastic optimal control models. Ann. Oper. Res 88, 183–197 (1999)
Rezaali, E., Ghane, F.H., Ebadizadeh, H.: Syndetically transitive and syndetically sensitive iterated function systems. J. Dyn. Syst. Geom. Theor. 16(2), 129–137 (2018)
Shao, S.: Proximity and distality via Furstenberg families. Topol. Appl. 153(12), 2055–2072 (2006)
Tan, F., Xiong, J.: Chaos via Furstenberg family couple. Topol. Appl. 156(3), 525–532 (2009)
Thakur, R., Das, R.: Devaney chaos and stronger forms of sensitivity on the product of semiflows. Semigroup Forum 98(3), 631–644 (2019)
Thakur, R., Das, R.: Stronger forms of sensitivity on product dynamical system via Furstenberg families. Studia Sci. Math. Hungar. 56(4), 440–453 (2019)
Vasisht, R., Das, R.: Furstenberg families and transitivity in non-autonomous systems. Asian Eur. J. Math. 13, 1 (2020). 2050029, 16
Wang, H., Xiong, J., Tan, F.: Furstenberg families and sensitivity. Discrete Dyn. Nat. Soc., (2010). https://doi.org/10.1155/2010/649348
Wu, X., Li, R., Zhang, Y.: The multi-F-sensitivity and \(( F_1, F_2)\)-sensitivity for product systems. J. Nonlinear Sci. Appl. 9(6), 4364–4370 (2016)
Wu, X., Wang, J., Chen, G.: F-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429(1), 16–26 (2015)
Xiong, J., Fu, H., Wang, H.: A class of Furstenberg families and their applications to chaotic dynamics. Sci. China Math. 57(4), 823–836 (2014)
Acknowledgements
Authors are thankful to the referees for their valuable comments and suggestions for improvement of the paper. The first author is supported by CSIR-SRF Sr.No. 1121641416 Ref.No: 18/12/2016(ii) EU-V (File No: 09/045(1532)/2017-EMR-I) for carrying out this research work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Thakur, R., Das, R. Transitivity and sensitivity of iterated function systems via Furstenberg families. Aequat. Math. 94, 1123–1140 (2020). https://doi.org/10.1007/s00010-020-00757-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-020-00757-8