Abstract
The Kolmogorov-Sinai entropy was introduced in connection with the problem of the isomorphism of dynamical systems. Recall that a dynamical system is a quadruplet (Ω, ℒ, P, T), where (Ω, ℒ, P) is a probability space and T: Ω → Ω is a measure preserving map, i.e., T -1(A) ∈ ℒ and P(T -1(A)) = P(A), whenever A ∈ ℒ.
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The notion of the entropy of dynamical systems was introduced by Kolmogorov [2] and Sinai [1]. With the help of this notion the existence of non-isomorphic Bernoulli schemes was proved. (The opposite problem has been solved 10 years latter, Ornstein [1]: if two Bernoulli schemas have the same entropy then they are isomorphic.)
The success of the Kolmogorov-Sinai entropy stimulated the introduction an analogous construction in other structures. The most famous is the notion of topological entropy, Adler, Konheim and McAndrew [1]. A relation between the topological entropy and the metric entropy was discovered by Goodwyn [1], and is described in the book Walters [1] (see also Komorník and Komorníková [2]). The probability structure was omitted in Kluvánek and Riečan [1]. There are at least two unifying theories: Riečan [15], Palm [1]. This was continued in the papers Grošek [1] and Komorníková [1]. It is interesting that the idea of topological entropy appeared five years ago in the paper by Winkelbauer [2].
A generalization of the notion of entropy is the notion of sequential entropy (A-entropy). In its definition (Kušnirenko [1]) not all natural numbers enter, but only those belonging to A. Sequential entropy can distinguish dynamical systems that are not distinguishable by the Kolmogorov and Sinaj entropy. A topological analogue of A-entropy was given by Newton [1]. In the papers Komorník and Komorníková [1], [3] there are improved some results by Newton [1] and Goodwyn [1] about the relation between entropy and A-entropy, or A-entropy and topological A-entropy.
A similar position as A-entropy has the fuzzy entropy studied in the chapter. Material presented in the chapter has been published by Du-mitrescu [1–7], Hudetz [1–5], Maličký and Riečan [1], Markechová [1–8], Markechová and Riečan [1], [2] (see also Mesiar [7]), Riečan [34], [36], [37], Rybárik [1].
The notion of g-entropy was inspired by the g-calculus developed by Pap and his school (Pap [7], Ralevic [1], Mesiar and Rybárik [1], [2], Marková and Riečan [1]).
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© 1997 Beloslav Riecan and Tibor Neubrunn
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Riečan, B., Neubrunn, T. (1997). The entropy of fuzzy dynamical systems. In: Integral, Measure, and Ordering. Mathematics and Its Applications, vol 411. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8919-2_10
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DOI: https://doi.org/10.1007/978-94-015-8919-2_10
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4855-4
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