Previous |  Up |  Next

Article

Keywords:
weakly fuzzy compact; weakly fuzzy compact topological dynamical system; weakly fuzzy topological entropy
Summary:
In 2005, İ. Tok fuzzified the notion of the topological entropy R. A. Adler et al. (1965) using the notion of fuzzy compactness of C. L. Chang (1968). In the present paper, we have proposed a new definition of the fuzzy topological entropy of fuzzy continuous mapping, namely weakly fuzzy topological entropy based on the notion of weak fuzzy compactness due to R. Lowen (1976) along with its several properties. We have shown that the topological entropy R. A. Adler et al. (1965) of continuous mapping $\psi \colon (X,\tau )\rightarrow (X,\tau )$, where $(X,\tau )$ is compact, is equal to the weakly fuzzy topological entropy of $\psi \colon (X,\omega (\tau ))\rightarrow (X,\omega (\tau ))$. We have also established an example that shows that the fuzzy topological entropy of İ. Tok (2005) cannot give such a bridge result to the topological entropy of Adler et al. (1965). Moreover, our definition of the weakly fuzzy topological entropy can be applied to find the topological entropy (namely weakly fuzzy topological entropy $h_w(\psi )$) of the mapping $\psi \colon X\rightarrow X$ (where $X$ is either compact or weakly fuzzy compact), whereas the topological entropy $h_a(\psi )$ of Adler does not exist for the mapping $\psi \colon X\rightarrow X$ (where $X$ is non-compact weakly fuzzy compact). Finally, a product theorem for the weakly fuzzy topological entropy has been established.
References:
[1] Adler, R. L., Konheim, A. G., McAndrew, M. H.: Topological entropy. Trans. Am. Math. Soc. 114 (1965), 309-319. DOI 10.1090/S0002-9947-1965-0175106-9 | MR 175106 | Zbl 0127.13102
[2] Afsan, B M U., Basu, C. K.: Fuzzy toplogical entropy of fuzzy continuous functions on fuzzy topological spaces. Appl. Math. Lett. 24 (2011), 2030-2033. DOI 10.1016/j.aml.2011.05.037 | MR 2826120 | Zbl 1269.54003
[3] Bowen, R.: Topological entropy and axiom $A$. Global Analysis, Proc. Sympos. Pure Math. 14 (1970), 23-42. MR 0262459 | Zbl 0207.54402
[4] Bowen, R.: Topological entropy for noncompact sets. Trans. Am. Math. Soc. 184 (1973), 125-136. DOI 10.1090/S0002-9947-1973-0338317-X | MR 0338317 | Zbl 0274.54030
[5] Cánovas, J. S., Kupka, J.: Topological entropy of fuzzified dynamical systems. Fuzzy Sets Syst. 165 (2011), 37-49. DOI 10.1016/j.fss.2010.10.020 | MR 2754584 | Zbl 1252.37018
[6] Cánovas, J. S., Rodríguez, J. M.: Topological entropy of maps on the real line. Topology Appl. 153 (2005), 735-746. DOI 10.1016/j.topol.2005.01.006 | MR 2201485 | Zbl 1085.37010
[7] Pe{ñ}a, J. S. Cánovas, López, G. Soler: Topological entropy for induced hyperspace maps. Chaos Solitons Fractals 28 (2006), 979-982. DOI 10.1016/j.chaos.2005.08.173 | MR 2212786 | Zbl 1097.54036
[8] Chang, C. L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24 (1968), 182-190. DOI 10.1016/0022-247X(68)90057-7 | MR 0236859 | Zbl 0167.51001
[9] Dumitrescu, D.: On fuzzy partitions. Prepr., "Babes-Bolyai" Univ., Fac. Math., Res. Semin. 2 (1983), 57-60. MR 0750491 | Zbl 0529.62048
[10] Dumitrescu, D.: Entropy of fuzzy process. Fuzzy Sets Syst. 55 (1993), 169-177. DOI 10.1016/0165-0114(93)90129-6 | MR 1215137 | Zbl 0818.28008
[11] Dumitrescu, D.: Fuzzy measure and the entropy of fuzzy partitions. J. Math. Anal. Appl. 176 (1993), 359-373. DOI 10.1006/jmaa.1993.1220 | MR 1224152 | Zbl 0782.28012
[12] Dumitrescu, D.: Entropy of a fuzzy dynamical systems. Fuzzy Sets Syst. 70 (1995), 45-57. DOI 10.1016/0165-0114(94)00245-3 | MR 1323286 | Zbl 0876.28029
[13] Dumitrescu, D., Barbu, M.: Fuzzy entropy and processes. Prepr., "Babes-Bolyai" Univ., Fac. Math., Res. Semin. 6 (1985), 71-74. MR 0842212
[14] Goodwyn, L. W.: Topological entropy bounds measure-theoretic entropy. Proc. Am. Math. Soc. 23 (1969), 679-688. DOI 10.1090/S0002-9939-1969-0247030-3 | MR 0247030 | Zbl 0186.09804
[15] Goodwyn, L. W.: Comparing topological entropy and measure-theoretic entropy. Am. J. Math. 94 (1972), 366-388. DOI 10.2307/2374626 | MR 0310191 | Zbl 0249.54021
[16] Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fractals 33 (2007), 76-86. DOI 10.1016/j.chaos.2005.12.033 | MR 2301847 | Zbl 1152.37306
[17] Liu, L., Wang, Y., Wei, G.: Topological entropy of continuous functions on topological spaces. Chaos Solitons Fractals 39 (2009), 417-427. DOI 10.1016/j.chaos.2007.04.008 | MR 2504577 | Zbl 1197.37015
[18] Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56 (1976), 621-633. DOI 10.1016/0022-247X(76)90029-9 | MR 0440482 | Zbl 0342.54003
[19] Lowen, R.: Initial and final fuzzy topologies and the fuzzy Tychonoff theorem. J. Math. Anal. Appl. 58 (1977), 11-21. DOI 10.1016/0022-247X(77)90223-2 | MR 0440483 | Zbl 0347.54002
[20] Markechová, D.: The entropy of fuzzy dynamical systems. BUSEFAL 38 (1989), 38-41.
[21] Markechová, D.: Isomorphism and conjugation of fuzzy dynamical systems. BUSEFAL 38 (1989), 94-101. Zbl 0677.93033
[22] Markechová, D.: The entropy of fuzzy dynamical systems and generators. Fuzzy Sets Syst. 48 (1992), 351-363. DOI 10.1016/0165-0114(92)90350-D | MR 1178175 | Zbl 0754.60005
[23] Markechová, D.: A note to the Kolmogorov-Sinaj entropy of fuzzy dynamical systems. Fuzzy Sets Syst. 64 (1994), 87-90. DOI 10.1016/0165-0114(94)90009-4 | MR 1281288 | Zbl 0845.93054
[24] Pu, P.-M., Liu, Y.-M.: Fuzzy topology. I: Neighborhood structure of a fuzzy point and Moore-Smith convergence. J. Math. Anal. Appl. 76 (1980), 571-599. DOI 10.1016/0022-247X(80)90048-7 | MR 0587361 | Zbl 0447.54006
[25] Pu, P.-M., Liu, Y.-M.: Fuzzy topology. II: Product and quotient spaces. J. Math. Anal. Appl. 77 (1980), 20-37. DOI 10.1016/0022-247X(80)90258-9 | MR 0591259 | Zbl 0447.54007
[26] Riečan, B., Markechová, D.: The entropy of fuzzy dynamical systems, general scheme and generators. Fuzzy Sets Syst. 96 (1998), 191-199. DOI 10.1016/S0165-0114(96)00266-7 | MR 1614814 | Zbl 0926.94012
[27] Thomas, R. F.: Some fundamental properties of continuous functions and topological entropy. Pac. J. Math. 141 (1990), 391-400. DOI 10.2140/pjm.1990.141.391 | MR 1035451 | Zbl 0661.58026
[28] Tok, I.: On the fuzzy topological entropy function. JFS 28 (2005), 74-80.
[29] Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79. Springer, New York (1982). DOI 10.1007/978-1-4612-5775-2 | MR 0648108 | Zbl 0475.28009
[30] Zadeh, L. A.: Fuzzy sets. Inf. Control 8 (1965), 338-353. DOI 10.1016/S0019-9958(65)90241-X | MR 0219427 | Zbl 0139.24606
Partner of
EuDML logo