# zbMATH — the first resource for mathematics

Sufficient families and entropy of inverse limit. (English) Zbl 0956.37005
Summary: The purpose of the present paper is to study the metric entropy $$h(\phi , \mathcal N)$$ of an $$F$$-measure preserving transformation $$\phi$$ relative to a $$\sigma$$-algebra $$\mathcal N$$ of an $$F$$-dynamical system. Concepts of sufficient families and generators are introduced and a few results are proved. Finally, the entropy of the inverse limit of an inverse spectrum of $$F$$-dynamical systems is obtained.

##### MSC:
 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A50 Dynamical systems and their relations with probability theory and stochastic processes
Full Text:
##### References:
 [1] BROWN J. R.: Ergodic Theory and Topological Dynamics. Academic Press, Inc., London, 1976. · Zbl 0334.28011 [2] BUTNARIU D.: Additive fuzzy measures and integrals. J. Math. Anal. Appl. 93 (1983), 436-452. · Zbl 0516.28006 [3] DUGUNDJI J.: Topology. Prentice Hall of India Pvt. Ltd., New Delhi, 1975. · Zbl 0144.21501 [4] DUMITRESCU D.: Fuzzy measures and the entropy of fuzzy partitions. J. Math. Anal. Appl. 176 (1993), 359-373. · Zbl 0782.28012 [5] HALMOS P. R.: Measure Theory. Van Nostrand Reinhold, Princeton, NJ, 1950. · Zbl 0040.16802 [6] KATOK A.-HASSELBLATT B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995. · Zbl 0878.58020 [7] KHARE M.: Fuzzy \sigma -algebras and conditional entropy. Fuzzy Sets and Systems 102 (1999), 287-292. · Zbl 0935.28011 [8] KLEMENT E. P.: Fuzzy \sigma -algebras and fuzzy measurable functions. Fuzzy Sets and Systems 4 (1980), 83-93. · Zbl 0444.28001 [9] MALIČKÝ P.-RIECAN B.: On the entropy of dynamical systems. Proc. Conf. Ergodic Theory and Related Topics II (Georgenthal 1986), Teubner, Leipzig, 1987, pp.135-138. [10] MARKECHOVÁ D.: The entropy of fuzzy dynamical systems and generators. Fuzzy Sets and Systems 48 (1992), 351-363. · Zbl 0754.60005 [11] MARKECHOVÁ D.: Entropy of complete fuzzy partitions. Math. Slovaca 43 (1993), 1-10. · Zbl 0772.94002 [12] MESIAR R.: The Bayes principle and the entropy on fuzzy probability spaces. Internat. J. Gen. Systems 20 (1991), 67-72. · Zbl 0737.60004 [13] PIASECKI K.: Probability of fuzzy events defined as denumerable additive measure. Fuzzy Sets and Systems 17 (1985), 271-284. · Zbl 0604.60005 [14] RIEČAN B.: A new approach to some notions of statistical quantum mechanics. Busefal 35 (1988), 4-6. [15] RIEČAN B.-DVURECENSKIJ A.: On randomness and fuzziness. Progress in Fuzzy Sets in Europe, 1986, Polska Akademia Nauk, Warszawa, 1988, pp. 321-326. [16] RIEČAN B.-NEUBRUNN T.: Integral, Measure and Ordering. Kluwer Acad. Publ.; Ister Press, Dordrecht; Bratislava, 1997. · Zbl 0916.28001 [17] SRIVASTAVA P.-KHARE M.: Conditional entropy and Rokhlin metric. Math. Slovaca (1999), 433-441. · Zbl 0949.28015 [18] SRIVASTAVA P.-KHARE M.-SRIVASTAVA Y.K.: A fuzzy measure algebra as a metric space. Fuzzy Sets and Systems 79 (1996), 395-400. · Zbl 0872.28012 [19] SRIVASTAVA P.-KHARE M.-SRIVASTAVA Y. K.: m-equivalence, entropy and F-dynamical systems. Fuzzy Sets and Systems · Zbl 0982.37006 [20] SRIVASTAVA P.-KHARE M.-SRIVASTAVA Y. K.: Fuzzy dynamical systems - inverse and direct spectra. Fuzzy Sets and Systems · Zbl 0974.37011 [21] SRIVASTAVA Y. K.: Fuzzy Probability Measures and Dynamical Systems. D. Phil. Thesis, Allahabad University, 1993.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.