Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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TOPOLOGICAL ENTROPY OF ONE DIMENSIONAL ITERATED FUNCTION SYSTEMS

  • Received : 2020.02.29
  • Accepted : 2020.09.30
  • Published : 2020.12.25
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Abstract

In this paper, topological entropy of iterated function systems (IFS) on one dimensional spaces is considered. Estimation of an upper bound of topological entropy of piecewise monotone IFS is obtained by open covers. Then, we provide a way to calculate topological entropy of piecewise monotone IFS. In the following, some examples are given to illustrate our theoretical results. Finally, we have a discussion about the possible applications of these examples in various sciences.

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References

  1. R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc, 114 (1965), 309-319. https://doi.org/10.1090/S0002-9947-1965-0175106-9
  2. AlsedA, LluAs, J. Llibre, and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, World Scientific Publishing Company (2000).
  3. L. Alsed'a, S. Kolyada, J. Llibre and L. Snoha, Axiomatic Definition of the Topological Entropy on the Interval, Aequationes Math 65, 1-2 (2003), 113-132. https://doi.org/10.1007/s000100300008
  4. M.F. Barnsley and A. Vince, The Conley attractor of an iterated function system, Bull. Aust. Math. Soc, 88 (2013), 267-279. https://doi.org/10.1017/S0004972713000348
  5. J. Baez, classical Mechanics, Lecture, 14 (February 26, 2008).
  6. Berryman and A. Alan, The orgins and evolution of predator-prey theory, Ecology, 73.5 (1992), 1530-1535. https://doi.org/10.2307/1940005
  7. R. Bowen, Entropy for group endomorphism and homogeneous space, Trans. Amer. Math. Soc, 153 (1971), 401-414. https://doi.org/10.2307/1995565
  8. Ch. Corda, M. Fatehi Nia, M. R. Molaei and Y. Sayyari, Entropy of iterated function systems and their relations with black holes and bohr-like black holes entropies, Entropy, 20 (2018), 56-72. https://doi.org/10.3390/e20010056
  9. Cuomo, M. Kevin and Alan V. Oppenheim, Chaotic signals and systems for communications, IEEE International Conference on Acoustics, (1993).
  10. X. Dai, Z. Zhou and X. Geng., Some Relations between Hausdorff-dimensions and Entropies, J. Sci. China Ser. A, 41 (1998), 1068-1075. https://doi.org/10.1007/BF02871841
  11. E. I. Dinaburg, On the Relations Among Various Entropy Characteristics of Dynamical Systems, Math. USSR Izv, 5 (1971), 337-378. https://doi.org/10.1070/IM1971v005n02ABEH001050
  12. O. Garasym, J. P. Lozi, and R. Lozi, How useful randomness for cryptography can emerge from multicore-implemented complex networks of chaotic maps. J. Difference Equ, Appl, 23 (2017), 821-859. https://doi.org/10.1080/10236198.2017.1287176
  13. T. N. T. Goodman, Z. Jinlian and H. Lianfa, Relating Topological Entropy and Measure Entropy, Bull. Lond. Math, 43 (1971), 176-180.
  14. S. Ito, An Estimate from above for the Entropy and the Topological Entropy of a C1-diffeomorphism, Proc. Japan Acad, 46 (1970), 226-230. https://doi.org/10.3792/pja/1195520395
  15. S. Ito, On the Topological Entropy of a Dynamical System, Proc. Japan Acad, 4 (1969), 838-840. https://doi.org/10.3792/pja/1195520543
  16. Khalil. H, Nonlinear control, New York Pearson, (2015).
  17. M. J. Lee, Introduction to smooth manifolds, GTM, 218 (2002).
  18. M. Misiurewicz, W. Szlenk, Entropy of Piecewise Monotone Mappings, Asterisque, 50 (1978), 299-310.
  19. Nitecki and H. Zbigniew, Topological entropy and the preimage structure of maps, Real Anal. Exchange, 29 (2003/04), 9-41. https://doi.org/10.14321/realanalexch.29.1.0009
  20. Ogata, Katsuhiko, and Yanjuan Yang, Modern control engineering, Prentice-Hall(2002).
  21. M. Patrao, Entropy and its variational principle for noncompact metric space, Ergodic Theory and Dynamical Systems, 30 (2010), 1529-1542. https://doi.org/10.1017/S0143385709000674
  22. Thomson, William, Theory of vibration with applications, CrC Press. (2018).
  23. V. Volterra, Theory of functionals and of integral and integro-differential equations, Courier Corporation, 2005.
  24. Y. Zhu, Z. Jinlian and H. Lianfa , Topological entropy of a sequence of monotone maps on circles, J. Korean Math. Soc. 43 (2006), 373-382. https://doi.org/10.4134/JKMS.2006.43.2.373