Geodesic
Weakly fuzzy topological entropy
Mathematica Bohemica, Tome 147 (2022) no. 2, pp. 221-236
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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.
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.
DOI : 10.21136/MB.2021.0073-20
Classification : 37B40, 37B99, 54A40, 54C70
Keywords: weakly fuzzy compact; weakly fuzzy compact topological dynamical system; weakly fuzzy topological entropy 
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Afsan, B M Uzzal. Weakly fuzzy topological entropy. Mathematica Bohemica, Tome 147 (2022) no. 2, pp. 221-236. doi: 10.21136/MB.2021.0073-20

[1] Adler, R. L., Konheim, A. G., McAndrew, M. H.: Topological entropy. Trans. Am. Math. Soc. 114 (1965), 309-319. | DOI | MR | JFM

[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 | MR | JFM

[3] Bowen, R.: Topological entropy and axiom $A$. Global Analysis, Proc. Sympos. Pure Math. 14 (1970), 23-42. | MR | JFM

[4] Bowen, R.: Topological entropy for noncompact sets. Trans. Am. Math. Soc. 184 (1973), 125-136. | DOI | MR | JFM

[5] Cánovas, J. S., Kupka, J.: Topological entropy of fuzzified dynamical systems. Fuzzy Sets Syst. 165 (2011), 37-49. | DOI | MR | JFM

[6] Cánovas, J. S., Rodríguez, J. M.: Topological entropy of maps on the real line. Topology Appl. 153 (2005), 735-746. | DOI | MR | JFM

[7] Pe{ñ}a, J. S. Cánovas, López, G. Soler: Topological entropy for induced hyperspace maps. Chaos Solitons Fractals 28 (2006), 979-982. | DOI | MR | JFM

[8] Chang, C. L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24 (1968), 182-190. | DOI | MR | JFM

[9] Dumitrescu, D.: On fuzzy partitions. Prepr., "Babes-Bolyai" Univ., Fac. Math., Res. Semin. 2 (1983), 57-60. | MR | JFM

[10] Dumitrescu, D.: Entropy of fuzzy process. Fuzzy Sets Syst. 55 (1993), 169-177. | DOI | MR | JFM

[11] Dumitrescu, D.: Fuzzy measure and the entropy of fuzzy partitions. J. Math. Anal. Appl. 176 (1993), 359-373. | DOI | MR | JFM

[12] Dumitrescu, D.: Entropy of a fuzzy dynamical systems. Fuzzy Sets Syst. 70 (1995), 45-57. | DOI | MR | JFM

[13] Dumitrescu, D., Barbu, M.: Fuzzy entropy and processes. Prepr., "Babes-Bolyai" Univ., Fac. Math., Res. Semin. 6 (1985), 71-74. | MR

[14] Goodwyn, L. W.: Topological entropy bounds measure-theoretic entropy. Proc. Am. Math. Soc. 23 (1969), 679-688. | DOI | MR | JFM

[15] Goodwyn, L. W.: Comparing topological entropy and measure-theoretic entropy. Am. J. Math. 94 (1972), 366-388. | DOI | MR | JFM

[16] Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fractals 33 (2007), 76-86. | DOI | MR | JFM

[17] Liu, L., Wang, Y., Wei, G.: Topological entropy of continuous functions on topological spaces. Chaos Solitons Fractals 39 (2009), 417-427. | DOI | MR | JFM

[18] Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56 (1976), 621-633. | DOI | MR | JFM

[19] Lowen, R.: Initial and final fuzzy topologies and the fuzzy Tychonoff theorem. J. Math. Anal. Appl. 58 (1977), 11-21. | DOI | MR | JFM

[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. | JFM

[22] Markechová, D.: The entropy of fuzzy dynamical systems and generators. Fuzzy Sets Syst. 48 (1992), 351-363. | DOI | MR | JFM

[23] Markechová, D.: A note to the Kolmogorov-Sinaj entropy of fuzzy dynamical systems. Fuzzy Sets Syst. 64 (1994), 87-90. | DOI | MR | JFM

[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 | MR | JFM

[25] Pu, P.-M., Liu, Y.-M.: Fuzzy topology. II: Product and quotient spaces. J. Math. Anal. Appl. 77 (1980), 20-37. | DOI | MR | JFM

[26] Riečan, B., Markechová, D.: The entropy of fuzzy dynamical systems, general scheme and generators. Fuzzy Sets Syst. 96 (1998), 191-199. | DOI | MR | JFM

[27] Thomas, R. F.: Some fundamental properties of continuous functions and topological entropy. Pac. J. Math. 141 (1990), 391-400. | DOI | MR | JFM

[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 | MR | JFM

[30] Zadeh, L. A.: Fuzzy sets. Inf. Control 8 (1965), 338-353. | DOI | MR | JFM

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