The Notion of Topological Entropy in Fuzzy Metric Spaces

Abstract

The aim of this paper is to extend the notion of topological entropy for fuzzy semidynamical systems created by a self-map on a fuzzy metric space. We show that if a metric space has two uniformly equivalent metrics, then fuzzy entropy is a constant up to these two metrics. We present a method to construct chaotic fuzzy semidynamical systems with arbitrary large fuzzy entropy. We also prove that fuzzy entropy is a persistent object under a fuzzy uniformly topological equivalent relation.

Keywords

• Fuzzy entropy,fuzzy metric space,semicompact,fuzzy semicompact

References

1. [1] D. Bala, Geometric methods in study of the stability of some dynamical systems, Analele S¸tiintifice ale Universitatii “Ovidius” Constanta. Seria: Matematica 17 (2009), 27–35.
2. [2] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society 153 (1971), 401–414.
3. [3] M. Ciklov´a, Dynamical systems generated by functions with connected Gδ graphs, Real Analysis Exchange 30 (2004), 617–638.
4. [4] E. D’Aniello and U. B. Darji, Chaos among self-maps of the Cantor space, Journal of Mathematical Analysis and Applications 381 (2011), 781–788.
5. [5] E. I. Dinaburg, The relation between topological entropy and metric entropy, Doklady Akademii Nauk SSSR 190 (1970), 19–22.
6. [6] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395–399.
7. [7] A. George and P. Veeramani, Some theorems in fuzzy metric spaces, Journal of Fuzzy Mathematics 3 (1995), 933–940.
8. [8] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (1997), 365-368.

9. [9] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (2000), 485–489.
10. [10] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984), 215– 229.
11. [11] B. Kılı¸c and E. Ba¸s, Complex solutions for the Fisher equation and the Benjamin-BonaMahony equation, C¸ ankaya University Journal of Science and Engineering 7 (2010), 87–93.
12. [12] Z. Koˇcan, V. Korneck´a-Kurkov´a and M. M´alek, Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites, Ergodic Theory and Dynamical Systems 31 (2011), 165–175.
13. [13] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975), 336–344.
14. [14] H. Molaei and M.R. Molaei, Dynamically defined topological entropy, Journal of Dynamical Systems and Geometric Theories 6 (2008), 95–100.
15. [15] M. R. Molaei, The concept of synchronization from the observer’s viewpoint, C¸ ankaya University Journal of Science and Engineering 8 (2011), 255–262.
16. [16] M. Patrao, Entropy and its variational principle for non-compact metric spaces, Ergodic Theory and Dynamical Systems 30 (2010), 1529–1542.
17. [17] P. Walter, An Introduction to Ergodic Theory, Springer-Verlag, 1982.