m-generators of Fuzzy Dynamical Systems

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1 Çankaya University Journal of Science and Engineering Volume 9 (202), No. 2, m-generators of Fuzzy Dynamical Systems Mohammad Ebrahimi, and Uosef Mohamadi Department of Mathematics, Shahid Bahonar University of Kerman, , Kerman, Iran Corresponding author: mohamad ebrahimi@mail.uk.ac.ir Özet. Bu makalede, sonlu atomlu bir alt-σ-cebirine göre bulanık ölçüm koruyan dönüşümün entropisinin afin olduğunu ispatlıyor, daha sonra bir sonlu alt-σ-cebirinin entropisini hesaplama yöntemini sayılabilir çoklukta atomlu alt-σ-cebirine uygulanacak şekille genelleştiriyor, ve bulanık olasılık dinamik sistemlerin ergodik özelliklerini araştırıyoruz. Son olarak, bu kavram kullanılarak Kolmogorov-Sinai önermesinin [6, 9, 0] bir çeşidi veriliyor. Anahtar Kelimeler. Bulanık olasılık uzayı, entropi, bulanık dinamik sistemler, m- denklik, m-saflaştırma, bulanık m-üreteç. Abstract. In this paper we prove that the entropy of a fuzzy measure preserving transformation with respect to a sub-σ-algebra having finite atoms is affine and then we extend the method of computing the entropy of a finite sub-σ-algebra to a sub-σ-algebra having countable atoms, and we investigate the ergodic properties of fuzzy probability dynamical systems. At the end by using this notion, a version of Kolmogorov-Sinai proposition [6, 9, 0] is given. Keywords. Fuzzy probability space, entropy, fuzzy dynamical systems, m-equivalence, m-refinement, fuzzy m-generator.. Introduction and Preliminaries The main idea of fuzzy entropy is the substitution of partitions by fuzzy partitions. In some previous papers [, 2, 3] entropy of a fuzzy dynamical system has been defined. Also the notions of m-refinements and m-equivalence have been defined in [8]. In this paper we give a definition for the entropy of a sub-σ-algebra with countable atoms. Received April 7, 202; accepted February 20, 203. Türkçe özet ve anahtar kelimeler, orijinal İngilizce metindeki ilgili kısmın doğrudan tercümesi olup Çankaya University Journal of Science and Engineering editörlüğü tarafından yazılmıştır. Turkish abstract and the keywords are written by the editorial staff of Çankaya University Journal of Science and Engineering which are the direct translations of the related original English text. ISSN Çankaya University

2 68 Ebrahimi and Mohamadi 2. Fuzzy Dynamical Systems We recall that a fuzzy set in a nonempty set X is an element of the family I X of all functions from X to closed unit interval I = [0, ]. A sequence {λ i } of fuzzy sets in X increase to λ I X (written as λ i λ) if {λ i (x)} i= is monotonic increasing and converges to λ(x) for each x in X. A fuzzy σ-algebra M on a non-empty set X is a subset of I X which satisfies the following conditions: (i) M, (ii) λ M λ M, (iii) if {λ i } i= is a sequence in M then i= λ i = sup i λ i M. If N and N 2 are fuzzy σ-algebras on X then N N 2 is the smallest fuzzy σ-algebra that contains N N 2, denoted by [N N 2 ]. A fuzzy probability measure m over M is a function m : M I which fulfills the conditions: (i) m() =, (ii) m( λ) = m(λ), (iii) m(λ µ) + m(λ µ) = m(λ) + m(µ) for each λ, µ M, (iv) for each sequence {λ i } i= in M such that λ i λ, m(λ) = sup i m(λ i ). The triple (X, M, m) is called a fuzzy probability measure space and the elements of M are called fuzzy measurable sets [8]. Definition 2.. Let (X, M, m) be a fuzzy probability measure space, the elements µ, λ of M are called m-disjoint if m(λ µ) = 0. A relation = (mod m) on M is defined as follows; λ = µ (mod m) iff m(λ) = m(µ) = m(λ µ), λ, µ M. Relation = (mod m) is an equivalence relation. M denotes the set of all equivalence classes induced by this relation, and µ is the equivalence class determined by µ. For λ, µ M, λ µ = 0 (mod m) iff λ, µ are m-disjoint. We shall identify µ with µ [8]. Definition 2.2. Let (X, M, m) be a fuzzy probability measure space, and N be a fuzzy sub-σ-algebra of M. Then an element µ Ñ is an atom of N if (i) m(µ) > 0, (ii) for each λ Ñ such that m(λ µ) = m(λ) m(µ) then m(λ) = 0, [8].

3 CUJSE 9 (202), No Proposition 2.3. Let (X, M, m) be a fuzzy probability measure space, and N be a fuzzy sub-σ-algebra of M. If µ, µ 2 are disjoint atoms of N then they are m-disjoint. Proof. See [8]. The set of all atoms of N is denoted by N. We define F (M) as below F (M) = {N : N is a sub-σ-algebra of M with finite atoms}. Definition 2.4. Suppose (X, M, m) and (Y, N, n) are fuzzy probability measure spaces. A transformation ϕ : (X, M, m) (Y, N, n) is said to be a fuzzy measure preserving if (i) ϕ (µ) M for every µ N, (ii) m(ϕ (µ)) = n(µ) for all µ N. Definition 2.5. A fuzzy dynamical system is denoted by (X, M, m, ϕ) where (X, M, m) is a fuzzy probability measure space and ϕ is a fuzzy measure preserving transformation. Definition 2.6. The entropy of N F (M) is given by H(N, m) = m(µ)log m(µ), µ N and the mean entropy of ϕ on N of the fuzzy dynamical system (X, M, m, ϕ) is defined by n h(n, M, ϕ) = lim n n H( ϕ i (N), m). Note that ϕ i (N) = {ϕ i (µ) : µ N} is an element of F (M) and n i= ϕ i (N) is the smallest fuzzy σ-algebra containing n ϕ i (N), and the above limit exists [8]. Proposition 2.7. The mean entropy of ϕ on N of the fuzzy dynamical system (X, M, m, ϕ) is affine, i.e, h(n, λm + ( λ)m 2, ϕ) = λh(n, m, ϕ) + ( λ)h(n, m 2, ϕ), for each pair m and m 2 of fuzzy probability measures, N F (M) and λ [0, ]. Proof. If m and m 2 are two fuzzy probability measures and λ [0, ] then i= H(N, λm + ( λ)m 2 ) λh(n, m ) + ( λ)h(n, m 2 ). ()

4 70 Ebrahimi and Mohamadi The concavity inequality () is a direct consequence of the definition of H(N, m) and the concavity of the function x xlog x. Conversely, one has inequalities log(λm (µ i ) + ( λ)m 2 (µ i )) log λ log(m (µ i )), and log(λm (µ i ) + ( λ)m 2 (µ i )) log( λ) log(m 2 (µ i )), since x log x is decreasing. Therefore one obtains the convexity bound H(N, λm + ( λ)m 2 ) λh(n, m ) + ( λ)h(n, m 2 ) λ log λ ( λ) log ( λ). (2) Now replacing N by n ϕ i (N) in (), dividing by n and taking the lim n gives h(n, λm + ( λ)m 2, ϕ) λh(n, m, ϕ) + ( λ)h(n, m 2, ϕ). Similarly from (2), since (λ log λ + ( λ) log ( λ)) n one deduces the converse inequality 0 as n, h(n, λm + ( λ)m 2, ϕ) λh(n, m, ϕ) + ( λ)h(n, m 2, ϕ). Hence one concludes the map m h(n, m, ϕ) is affine. This is a somewhat surprising and is of great significance in the application of fuzzy mean entropy. 3. Ergodic Measures and Weak-Mixing Definition 3.. Given a fuzzy probability space (X, M, m), a fuzzy measure preserving transformation ϕ : X X is called ergodic if for every atom γ M with ϕ (γ) = γ we have that either m(γ) = 0 or m(γ) =. Alternatively we say that m is ϕ-ergodic. Proposition 3.2. Let Σ denote the set of fuzzy invariant probability measures on X. m Σ is ergodic if whenever there exists m, m 2 Σ and 0 < λ < with m = λm + ( λ)m 2 then m = m 2. Proof. If m is not ergodic then we can find γ M with ϕ (γ) = γ and 0 < m(γ) < but for every atom µ M we can write µ = (µ γ) (µ ( γ)).

5 CUJSE 9 (202), No. 2 7 Therefore m(µ) = m((µ γ) (µ ( γ))) ( ) ( ) m(µ γ) m(µ ( γ)) = m(γ) + m( γ) m(γ) m( γ) = λm (µ) + ( λ)m 2, where λ = m(γ) and m (µ) = m(µ γ)/m(γ), m 2 (µ) = m(µ ( γ))/m( γ) this shows that m = λm + ( λ)m 2 (µ). Definition 3.3. Let (X, M, m, ϕ) be a fuzzy dynamical system, we say that ϕ is weak-mixing if for any µ, λ M we have that, k k m(ϕ n (µ) λ) m(µ)m(λ) 0 as k. Proposition 3.4. If a transformation ϕ : X X on a fuzzy probability measure space (X, M, m) is weak-mixing then it is necessarily ergodic. Proof. If ϕ is weak-mixing then by definition we have that for any µ, λ M, k k m(ϕ n (µ) λ) m(µ)m(λ) 0 as k. By the triangle inequality we have that, k m(ϕ n (µ) λ) m(µ)m(λ) k k m(ϕ n (µ) λ) m(µ)m(λ) 0. k If we assume (for a contradiction) that ϕ was not ergodic then there would exist a ϕ-invariant atom γ M with ϕ (γ) = γ with 0 < m(γ) <. If we take µ = γ and λ = γ then since m(ϕ n (γ) ( γ)) = m(γ ( γ)), for all n 0, we deduce that m(γ) m( γ) = 0 giving the required contradiction. Thus ϕ is ergodic. 4. Entropy of a Sub-σ-Algebra with Countable Atoms In this section we introduce the notion of entropy of a sub-σ-algebra with countable atoms. We introduce F (M) as below, F (M) = {N : N is a sub-σ-algebra of M with countable atoms}. Assume that M is a σ-algebra and N, N 2 F (M), and {λ i : i N} and {µ j : j N} denote the atoms of N and N 2 respectively, then the atoms of N N 2

6 72 Ebrahimi and Mohamadi are λ i µ j which m(λ i µ j ) > 0 for each i, j N. If γ M we set N γ = {λ i γ : m(λ i γ) > 0, i N}. Proposition 4.. Let {λ i : i N} be an m-disjoint collection of fuzzy measurable sets of fuzzy probability measure space (X, M, m), then, ( ) m (λ i ) = m(λ i ). i= i= Proof. See [8]. Definition 4.2. Let (X, M, m) be a fuzzy probability measure space and N, N 2 F (M). We say that N 2 is an m-refinement of N, denoted by N m N 2, if for each µ N 2 there exists λ N such that m(λ µ) = m(µ). Proposition 4.3. Let (X, M, m) be a fuzzy probability measure space and N, N 2, N 3 F (M). If N m N 2 then, N N 3 m N 2 N 3. Proof. See [8]. Definition 4.4. Let (X, M, m) be a fuzzy probability space, and N be a sub-σalgebra of M for which N F (M). The entropy of N is defined as where {µ i : i N} are atoms of N. H(N) = log sup m(µ i ), i N Definition 4.5. Let (X, M, m) be a fuzzy probability measure space and N F (M). The conditional entropy of N given γ M is defined by where, H(N γ) = log sup m(µ i γ), i N m(µ i γ) = m(µ i γ) m(γ) (m(γ) 0). Proposition 4.6. Let (X, M, m) be a fuzzy probability measure space, and N, N 2 F (M) for which N = {λ i : i N} and N 2 = {µ j : j N}. Then, (i) N m N 2 H(N ) H(N 2 ), (ii) N m N 2 H(N γ) H(N 2 γ).

7 CUJSE 9 (202), No Proof. (i) Suppose N m N 2, and then for each µ j N 2 there exists λ ij that, m(µ j λ ij ) = m(µ j ) but λ ij µ j λ ij. Then, N such m(λ ij µ j ) m(λ ij ) m(µ j ) m(λ ij ) m(µ j ) sup λ ij N m(λ ij ). Since µ j is arbitrary we have: sup j N m(µ j ) sup i N m(λ i ) and then we have H(N ) H(N 2 ) since f(x) = log x is a decreasing function. (ii) Supppose N m N 2, by Proposition 4.3 we have, N γ m N 2 γ, and by (i) we conclude that H(N γ) H(N 2 γ) log sup i N sup j N m(λ i γ) log sup m(µ j γ) j N m(µ j γ) sup m(λ i γ) i N m(µ j γ) sup j N m(γ) log sup i N sup i N m(λ i γ) m(γ) H(N γ) H(N 2 γ). m(λ i γ) m(γ) m(µ j γ) log sup j N m(γ) Definition 4.7. Let (X, M, m) be a fuzzy probability measure space and N, N 2 F (M). We say that N and N 2 are m-equivalent, denoted by N m N 2, if (i) for each µ N 2, m(µ ( {λ : λ N })) = m(µ), (ii) for each λ N, m(λ ( {µ : µ N 2 })) = m(λ). Proposition 4.8. Let (X, M, m) be a fuzzy probability measure space, and N, N 2 F (M). Then, N m N 2 N m N N 2. Proof. Assume that, N = {λ i : i N}, N 2 = {µ j : j N}. We know that N N 2 = {λ i µ j : λ i N, µ j N 2, m(λ i µ j ) > 0}. If α = {(i, j) : V ij = λ i µ j N N 2 } then α = i N {(i, j) : j β i} where β i = {j : m(v ij ) > 0} and i N. Note that if j / β i then m(v ij ) = 0 we have V ij = ( V ij ) = (λ i ( µ j )). i,j N i N j β i i N j β i

8 74 Ebrahimi and Mohamadi Since the collections of {λ i : i N} and {µ j : j N} are m-disjoint, we have, m(λ k ( V ij )) = m(λ k ( λ i ( µ j ))) i,j N i N j β i = m(λ k ( µ j )) j β i = m(λ k ( µ j )) j β k = m( (λ k µ j )) j β k = j β k m( V kj ) = j N m( V kj ) = m(λ k ( j N µ j )) = m(λ k ). Proposition 4.9. Let (X, M, m) be a fuzzy probability measure space, and N, N 2 F (M). If N m N 2 then, H(N ) H(N N 2 ). Proof. Suppose N m N 2, by Proposition 4.8 we have N m N N 2. Now suppose that θ N N 2 then θ = λ i µ j where λ i N and µ j N 2. So for λ i N, m(θ) = M(θ λ i ) and therefore we have N m N N 2. Now use Proposition 4.6, (i). Definition 4.0. Let (X, M, m) be a fuzzy probability measure space and N F (M). The diameter of N is defined as follows diam N = sup m(λ i ). λ i N Definition 4.. Let (X, M, m) be a fuzzy probability measure space and N, C F (M), where N = {λ i : i N}, C = {γ k : k N}. The conditional entropy of

9 CUJSE 9 (202), No N given C is defined as H(N C) = log sup i N = log sup j N diam(λ i C) diam C diam(n µ j ). diam C Proposition 4.2. Let (X, M, m) be a fuzzy probability measure space, and N, C, D F (M). Then, (i) C m D H(N C) H(N D), (ii) H(N C) H(N C), (iii) N m C H(N D) H(C D). Proof. Suppose that N = {λ i : i N}, C = {µ j : j N} and D = {γ k : k N}. (i) Suppose that C m D, then we have, H(C N) H(D N) log sup m(λ i µ j ) log sup m(λ i γ k ) i N i,k N Note that 0 < diam C. (ii) Obvious. (iii) Suppose N m C, then we have, sup m(λ i µ j ) sup m(λ i γ k ) i,j N i,k N sup m(λ i µ j ) sup m(λ i γ k ) diam C i,j N i,k N H(N C) H(N D). H(N D) H(C D) log sup m(λ i γ k ) log sup m(γ k µ j ) i,k N k,j N sup m(γ k µ j ) sup m(λ i γ k ) k,j N i,k N m(γ k µ j ) sup k,j N diam D m(λ i γ k ) log sup i,k N diam D H(N D) H(C D). sup m(λ i γ k ) i,k N diam D m(γ k µ j ) log sup k,j N diam D

10 76 Ebrahimi and Mohamadi Proposition 4.3. Suppose (X, M, m) is a fuzzy probability measure space, and N, N 2, N 3 F (M). Then, H(N N 2 N 3 ) = H(N N 2 ) + H(N 2 N N 3 ). Proof. Suppose that N = {λ i : i N}, N 2 = {µ j : j N} and N 3 = {γ k : k N}. We know that, But we can write, m(λ i µ j γ k ) H(N N 2 N 3 ) = log sup. i,j,k N diam N 3 m(λ i µ j γ k ) sup k N m(γ k ) and therefore the proof is obvious. = m(λ i µ j γ k ) sup i,k N m(λ i γ k ) sup i,k N m(λ i γ k ), sup k N m(γ k ) 5. Entropy of a Measure Preserving Transformation Definition 5.. Suppose ϕ : X X is a fuzzy measure preserving transformation of the fuzzy probability measure space (X, M, m). If N F (M), we define the entropy of ϕ with respect to N as n h(ϕ, N) = lim n n H( ϕ i (N)). We say (X, M, m, ϕ) is a fuzzy dynamical system. It is of course necessary to establish that the limit above exists, but this is a consequence of subadditivity []. Proposition 5.2. Suppose ϕ : (X, M, m) (Y, N, n) is a fuzzy measure preserving transformation. Then for each L F (N) we have H(L) = H(ϕ (L)). Proof. Since ϕ is measure preserving, for all µ L, we have m(ϕ (µ)) = n(µ) H(ϕ (L)) = log sup m(ϕ (µ)) µ L = log sup n(µ) µ L = H(L).

11 CUJSE 9 (202), No Proposition 5.3. Let (X, M, m, ϕ) be a fuzzy dynamical system and N, F (M). Then, (i) N m C h(ϕ, N) h(ϕ, C), (ii) h(ϕ, ϕ (N)) = h(ϕ, N), (iii) h(ϕ, r ϕ i (N)) = h(ϕ, N) for every r, (iv) if N, N 2 F (M) such that N m N 2 then, C ϕ (N ) m ϕ (N 2 ). Proof. (i) Follows from Proposition 4.3 and Proposition 4.6, (i). (ii) Obvious. (iii) h(ϕ, i= ϕ i (N)) = lim n n n H( j=0 r ϕ j ( r+n 2 = lim n n H( ϕ i (N)) ϕ i (N))) = lim ( r + n 2 r+n 2 )( n n r + n 2 )H( ϕ i (N)) = h(ϕ, ϕ(n)). (iv) Let ϕ (µ) ϕ (N 2 ) such that µ N 2. Then, m(ϕ (µ) ( {ϕ (λ) : λ N })) = m(ϕ (µ ( {λ : λ N }) The proof of = n(µ ( {λ : λ N })) = n(µ) = m(ϕ (µ)). m(ϕ (λ) ( {ϕ (µ) : µ N 2 })) = m(ϕ (λ)), where ϕ (λ) ϕ (N ) is similar. 6. Entropy and m-isomorphic Dynamical Systems Definition 6.. Let (X, M, m, ϕ) be a fuzzy dynamical system and L F (M). Suppose [L] denotes the m-equivalence class induced by L. Then the entropy

12 78 Ebrahimi and Mohamadi h(ϕ, [L]) of ϕ on L is defined as h(ϕ, [L]) = sup h(ϕ, N). N [L] Definition 6.2. A fuzzy dynamical system φ = (X, M, m, ϕ ) is a factor of fuzzy dynamical system φ 2 = (X 2, M 2, m 2, ϕ 2 ) if there exists an onto fuzzy measure preserving transformation (called homomorphism) ψ : φ 2 φ such that, and for each µ M, ψ ϕ 2 = ϕ ψ, m (µ) = m 2 (ψ (µ)). Proposition 6.3. Let φ = (X, M, m, ϕ ) be a factor of fuzzy dynamical system φ 2 = (X 2, M 2, m 2, ϕ 2 ), then for each L F (M ), h(φ, [L]) h(φ 2, [ψ (L)]), where ψ : φ 2 φ is the corresponding homomorphism. Proof. Suppose that N [L]. Then by Proposition 5.4, H(N) = H(ψ (N)). Now, h(φ, N) = lim n n n H( φ i (N)) n = lim n n H(ψ ( φ i (N))) = lim n = lim n n n H( n n H( = h(φ 2, ψ (N)). ψ φ i (N)) φ i 2 ψ (N)) As N ranges over an m-equivalence class [L] in F (M ), ψ (N) ranges over a subset of the m-equivalence class [ψ (L)] in F (M 2 ). Definition 6.4. Two dynamical systems φ = (X, M, m, ϕ ) and φ 2 = (X 2, M 2, m 2, ϕ 2 ) are said to be m-isomorphic if there exists an invertible fuzzy measure preserving transformation ψ : φ φ 2 (i.e both ψ and ψ are fuzzy measure preserving transformations) such that, ψ ϕ = ϕ 2 ψ.

13 CUJSE 9 (202), No The mapping ψ is called m-isomorphism. Proposition 6.5. Suppose φ = (X, M, m, ϕ ) and φ 2 = (X 2, M 2, m 2, ϕ 2 ) are m-isomorphic dynamical systems and ϕ is an ergodic fuzzy transformation. Then ϕ 2 is also ergodic. Proof. Let µ M 2 ; ϕ 2 (µ) = µ. By definition there exists an invertible fuzzy measure preserving transformation ψ of φ onto φ 2 such that, But ψ (µ) = γ M, and, So we have ψ ϕ = ϕ 2 ψ. ϕ 2 (µ) = ϕ 2 (ψ(γ)) = ψ ϕ (µ) = ψ(γ). ϕ (γ) = γ m (γ) = 0 or m (ψ (µ)) = 0 or m 2 (µ) = 0 or.. Proposition 6.6. Let φ = (X, M, m, ϕ ) and φ 2 = (X 2, M 2, m 2, ϕ 2 ) be m- isomorphic dynamical systems and ϕ be weak mixing. Then ϕ 2 is also a weak mixing. Proof. Since ϕ is weak mixing then we have that for any µ, λ M, k lim m (ϕ n (µ) λ) m (µ)m 2 (λ) = 0. k k We prove that for any η, ν M 2 we have k lim m (ϕ n (η) ν) m (η)m 2 (ν) = 0. k k Since φ and φ 2 are m-isomorphic, there is an invertible fuzzy measure preserving transformation ψ such that ψ ϕ = ϕ 2 ψ we have ψ ϕ n 2 = ϕ n ψ.

14 80 Ebrahimi and Mohamadi Since ψ is surjective and measure preserving, ψ (η) M, ψ (ν) M. Suppose that ψ (η) = µ, ψ (ν) = λ, then, k lim m (ϕ n (µ) λ) m (µ)m (λ) k k k = lim m (ψ ((ϕ n 2 (η) ν) m (ψ (η))m (ψ (ν)) k k k = lim m 2 (ϕ n 2 (η) ν) m 2 (η)m 2 (ν) k k = 0. Proposition 6.7. Let φ and φ 2 be m-isomorphic dynamical systems. Then for each L F (M), h(ϕ, [L]) = h(ϕ 2, [ψ (L)]), where ψ : φ φ 2 is the corresponding m-isomorphism. In the other words h(ϕ, [L]) is m-isomorphism invariant. Proof. Follows from Proposition Entropy and m-generators of Fuzzy Dynamical Systems Definition 7.. The entropy of the fuzzy dynamical system (X, M, m, ϕ) is the number h(ϕ) defined by: h(ϕ) = sup h(ϕ, ξ), ξ where the supremum is taken over all sub-σ-algebras of M where ξ F (M). Definition 7.2. ξ F (M) is said to be a fuzzy m-generator of the fuzzy dynamical system (X, M, m, ϕ) if there exists an integer r > 0 such that, r η m ϕ i ξ, for each η F (M). Proposition 7.3. If ξ is a m-generator of the fuzzy dynamical system (X, M, m, ϕ) then, h(ϕ, η) h(ϕ, ξ),

15 CUJSE 9 (202), No. 2 8 for each η F (M). Proof. Let η F (M) be any arbitrary sub-σ-algebra of M. Since ξ, is an m- generator, η r m ϕ i ξ from Proposition 5.3, (iii), r h(ϕ, η) h(ϕ, ϕ i ξ) = h(ϕ, ξ). Now we can deduce the following version of Kolmogorov-Sinai proposition. Proposition 7.4. If ξ is an m-generator of fuzzy dynamical system (X, M, m, ϕ) then, h(ϕ) = h(ϕ, ξ). Proof. Obvious. 8. Concluding Remarks and Open Problems In this paper we investigate the ergodic properties of fuzzy dynamical systems using the concept of atoms in a fuzzy σ-algebra. In this respect we introduce the m- generators of fuzzy dynamical systems. We have to consider a slight modification of some previously defined notions. A fuzzy version of Kolmogorov-Sinai proposition concerning the entropy of fuzzy dynamical system is given. This proposition enables us to compute the entropy for a class of fuzzy systems. An interesting open problem is to establish a proposition on existence of m-generators having finite entropy. References [] D. Dumitrescu, C. Hăloiu and A. Dumitrescu, Generators of fuzzy dynamical systems, Fuzzy Sets and Systems 3 (2000), [2] D. Dumitrescu, Entropy of a fuzzy dynamical system, Fuzzy Sets and Systems 70 (995), [3] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems 55 (993), [4] M. Ebrahimi, Generators of probability dynamical systems, Differential Geometry-Dynamical Systems 8 (2006), [5] M. Ebrahimi and N. Mohamadi, The entropy function on an algebraic structure with infinite partition and m-preserving transformation generators, Applied Sciences 2 (200), [6] A. N. Kolmogorov, Entropy per unit time as a metric invariant of automorphism, Doklady of Russian Academy of Sciences 24 (959),

16 82 Ebrahimi and Mohamadi [7] P. Serivastava, M. Khare and Y. K. Srivastava, A fuzzy measure algebra on a metric space, Fuzzy Sets and Systems 79 (996), [8] P. Serivastava, M. Khare and Y. K. Srivastava, m-equivalence, entropy and F -dynamical systems, Fuzzy Sets and Systems 2 (200), [9] Ya. Sinai, On the notion of entropy of a dynamical system, Doklady ofrussian Academy of Sciences 24 (959), [0] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 982.