Kybernetika 51 no. 2, 231-245, 2015

A local approach to g-entropy

Mehdi RahimiDOI: 10.14736/kyb-2015-2-0231

Abstract:

In this paper, a local approach to the concept of $g$-entropy is presented. Applying the Choquet`s representation Theorem, the introduced concept is stated in terms of $g$-entropy.

Keywords:

fuzzy entropy, $g$-entropy, local entropy

Classification:

28D20, 28E10

paper.pdf

References:

  1. M. Brin and A. Katok: On local entropy in geometric dynamics. Springer-Verlag, New York 1983 (Lecture Notes in Mathematics 1007) (1983), pp. 30-38.   DOI:10.1007/bfb0061408
  2. D. Butnariu and E. P. Klement: Triangular norm-based measures and their Markov kernel representation. J. Math. Anal. Appl. 162 (1991), 111-143.   DOI:10.1016/0022-247x(91)90181-x
  3. D. Dumitrescu: Measure-preserving transformation and the entropy of a fuzzy partition. In: 13th Linz Seminar on Fuzzy Set Theory, Linz 1991, pp. 25-27.   CrossRef
  4. D. Dumitrescu: Fuzzy measures and the entropy of fuzzy partitions. J. Math. Anal. Appl. 176 (1993), 359-373.   DOI:10.1006/jmaa.1993.1220
  5. D. Dumitrescu: Entropy of a fuzzy process. Fuzzy Sets and Systems 55 (1993), 169-177.   DOI:10.1016/0165-0114(93)90129-6
  6. D. Dumitrescu: Entropy of fuzzy dynamical systems. Fuzzy Sets and Systems 70 (1995), 45-57.   DOI:10.1016/0165-0114(94)00245-3
  7. D. Markechová: The entropy on F-quantum spaces. Math. Slovaca 40 (1990), 177-190.   CrossRef
  8. D. Markechová: The entropy of fuzzy dynamical systems and generators. Fuzzy Sets and Systems 48 (1992), 351-363.   DOI:10.1016/0165-0114(92)90350-d
  9. D. Markechová: Entropy of complete fuzzy partitions. Math. Slovaca 43 (1993), 1, 1-10.   CrossRef
  10. D. Markechová: A note to the Kolmogorov-Sinaj entropy of fuzzy dynamical systems. Fuzzy Sets and Systems 64 (1994), 87-90.   DOI:10.1016/0165-0114(94)90009-4
  11. B. McMillan: The basic theorems of information theory. Ann. Math. Statist. 24 (1953), 196-219.   DOI:10.1214/aoms/1177729028
  12. R. Mesiar and J. Rybárik: Entropy of fuzzy partitions: A general model. Fuzzy Sets and Systems 99 (1998), 73-79.   DOI:10.1016/s0165-0114(97)00024-9
  13. M. Rahimi and A. Riazi: Entropy operator for continuous dynamical systems of finite topological entropy. Bull. Iranian Math. Soc. 38 (2012), 4, 883-892.   CrossRef
  14. M. Rahimi and A. Riazi: On local entropy of fuzzy partitions. Fuzzy Sets and Systems 234 (2014), 97-108.   DOI:10.1016/j.fss.2013.02.006
  15. B. Riečan: On a type of entropy of dynamical systems. Tatra Mountains Math. Publ. 1 (1992), 135-140.   CrossRef
  16. B. Riečan: On the $g$-entropy and its Hudetz correction. Kybernetika 38 (2002), 4, 493-500.   DOI:10.1016/j.physa.2007.06.047
  17. B. Riečan and D. Markechová: The entropy of fuzzy dynamical systems, general scheme and generators. Fuzzy Sets and Systems 96 (1998), 191-199.   DOI:10.1016/s0165-0114(96)00266-7
  18. B. Riečan and T. Neubrunn: Integral, Measure, and Ordering. Kluwer, Dordrecht and Ister, Bratislava 1997.   DOI:10.1007/978-94-015-8919-2
  19. J. Rybárik: The entropy of the Q-F-dynamical systems. Busefal 48 (1991), 24-26.   CrossRef
  20. J. Rybárik: The entropy based on pseudoarithmetical operations. Tatra Mountains Math. Publ. 6 (1995), 157-164.   CrossRef
  21. Ya. Pesin: Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 32 (1977), 54-114.   DOI:10.1070/rm1977v032n04abeh001639
  22. R. Phelps: Lectures on Choquet's Theorem. Van Nostrand, Princeton, N.J. 1966.   CrossRef
  23. D. Ruelle: An inequality for the entropy of differential maps. Bol. Soc. Bras. de Mat. 9 (1978), 83-87.   DOI:10.1007/bf02584795
  24. P. Walters: An Introduction to Ergodic Theory. Springer-Verlag, 1982.   CrossRef