The entropy of fuzzy dynamical systems and generators

Fuzzy Sets and Systems 48 (1992) 351-363 North-Holland

351

The entropy of fuzzy dynamical systems and generators Dagmar Markechov~i Department of Mathematics, Pedagogical Faculty, 949 01 Nitra, Saratovsktl 19, Czechoslovakia Received October 1990

Abstract: In this paper the entropy and conditional entropy of stochastical complete repartitions are defined. The represented measure satisfies all properties analogous to the properties of entropy of measurable repartitions in the crisp case. Further, the notion of entropy of fuzzy dynamical systems is defined. The above notion generalizes the notion of entropy of dynamical systems from the classical probability theory. Finally, the Kolmogorov-Sinaj theorem on generators is proved for the fuzzy case.

Keywords: Fuzzy probability space; entropy; fuzzy dynamical systems; generators.

1. Introduction In classical probability theory [1], probability spaces (X, 5P, P) are studied. In this paper we shall work with a fuzzy generalization of notion of a probability space. Soft fuzzy probability spaces (X, M, m) are considered in [2]. Piasecki there introduces the notion of a stochastical complete repartition of a space (X, M, m). In this paper the entropy and conditional entropy of stochastical complete repartitions are studied. The main properties of such quantities are stated. The connection with the classical cases is also mentioned. Further, we introduce the notion of entropy of fuzzy dynamical systems and we formulate the Kolmogorov-Sinaj theorem on generators for fuzzy dynamical systems. Another approach to the problem of a fuzzy generalization of Kolmogorov-Sinaj's entropy is given in [3, 4].

2. Basic definitions and facts Here we follow mainly [2]. A soft fuzzy probability space is a triplet (X, M, m), where X is a nonempty set; M c (0, 1) x is a collection of fuzzy sets satisfying: (C1) If l(x) = 1 for all x • X, then 1 • M. (C2) Iffn • M , n = 1, 2 , . . . , then V~=lfn := supfn • M . (C3) If f e M, then f ' : = 1 - f • M. (C4) If ½(x) = ½ for all x • X, then ½~ M. The mapping (C5)

m:M---~ (0, ~) fulfils the following conditions:

m(f v ( l - f ) ) = 1 for a l l f • M.

(C6) If {fn}n is a finite or infinite sequence of pairwise W-separated fuzzy subsets from M (i.e. f~ ~< 1 - f j whenever i 4:j), then m(V~=lfn) = E~=1 rn(fn). 0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved

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The partial ordering relation ~< is defined in the following way: for every f, g • M , f <~g iff f ( x ) <~g(x) for all x • X. Using the complementation ':f---~f' for every fuzzy subset f • M, we see that the complementation ' satisfies two conditions: (i) ( f ' ) ' = f f o r every f • M; (ii) iff<~g, then g' ~ O, by the equality

m(f

I g) -

m(f

^

m(g)

) '

is a fuzzy P-measure on M called conditional probability of f given g (see [2]).

Example 2.1. Let (X, fie, P) be a probability space in the sense of classical probability theory. Put M = {XA:A • fie}, where Zm is the characteristic function of the set A • fie. Let us define the mapping m :M---~ (0, 1) by m(zA) = P(A). Then the triplet (X, M, m) is a soft fuzzy probability space. One also can consider the following extension of Example 2.1:

Example 2.2. Let (X, fie, P) be a probability space in the classical sense. Consider the set M "= {f;f:X----~ (0, 1), f is an fie-measurable mapping such that P ( f • (1, 3))=0}. If we define the mapping m:M---~(0, 1) by the equality m ( f ) = P ( f > l ) , then it is easy to see that the triplet (X, M, m) is a soft fuzzy probability space.

Example2.3. Let X = (0, 1), f :X---~ X, f ( x ) = x for every x • X, M = {f, f ' , f v f ' , f A f ' , 0, 1}. If we define the mapping m :M---~ (0, 1) by the equalities m(1) = m ( f v f ' ) = 1, m(0) = m ( f A f ' ) = 0 and m ( f ) = m ( f ' ) = ½, then the triplet (X, M, m) is a soft fuzzy probability space. A fuzzy generalization of the notion of measurable partition of a space (X, fie, P) from classical probability theory is the notion of stochastical complete repartition [2]. A finite set ~ / = {ft . . . . . fn} of pairwise W-separated fuzzy subsets from M such that m(V~'=lf~)= 1 is called a stochastical complete repartition. Let ~ be the set of all stochastical complete repartitions of a space (X, M, rn).

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3. The entropy of stochastical complete repartitions Let any soft fuzzy probability space (X, M, m) be given. Each M • ~ represents in the sense of classical probability theory a random experiment with finite number of outcomes {f,;f//• M} with probability distribution pi=m(fi). This is a legitimate probability distribution, since pg~>0 and ~7=lPi= ~7_lm(fi)=m(VT=lfii)= 1. Therefore we can define the information of any experiment = {fl . . . . . f, } • ~ by Shannon's formula:

H,~(M)= - ~ F(m(fi)), where F" (0, oo)-->R, i=l

F(x)=

jxlogx

ifx >0,

to

ifx =0.

(3.1)

In the set ~ of all stochastical complete repartitions one can define the relation ~< as follows: for every M, ~ • ~, M ~< ~ iff for every g • ~ there exists f • M such that g ~
Example 3.1. Let (X, 5¢, P) be a probability space in the classical sense. Let us consider the soft fuzzy probability space (X, M, m) from Example 2.1. Then the system ~ contains all repartitions of the type {gm, . . . . . gak}, where A i • ~f (i = 1 . . . . . k ), Ai fq Aj = 0 (i ~ j) and P(U~-I Ai) = 1. The entropy of a repartition M = {gn,, . . . . ZA~} is the number k

k

H,,,(~I) = - ~ F(m(XA,)) = -- ~ F(P(Ai)), i=1

i=1

which is the Shannon entropy of a measurable repartition {A~ . . . . . A k , A k + l } of the space (X, 5¢, P), where Ak÷l = ((_Jki_lAi) c (A c denotes the complement of a set A • 50 is a set of measure zero.

Example 3.2. Let (X, M, m) be the soft fuzzy probability space from Example 2.3. Then only the set M = {f, f ' } is a stochastical complete repartition; it has the nonzero entropy Hm(M)= log 2. Theorem 3.1. The entropy Hm: ~--~ R has the following properties: (P1) Hm(M) >! 0 for every M • ~; (P2) if M, ~ • ~, M <- ~ , then Hm(M) <~H m ( ~ ) ; (P3) Hm(M) <~ Hm(M v ~ ) for every M, ~ • ~. Proof. Property (P1) is evident. Let M, ~ • ~ , M = {fl . . . . . fn}, ~ = {g~ . . . . . gk}, M <~~. Then for every gj • ~ there exists rio • M such that gj ~
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Put tr={(i,j); m(f/A&)>O, i = 1 . . . . . n, j = l . . . . . k} and fl={i;m(f/)>O, i = 1 . . . . . n} the above fact along with (C6) and (C9) implies

Hm(~) = --Z F(m(gj)) = j=l

F(m(gj A f/))

~ j=1 i=1

= - ~. m(f/h &)log m(f/A gj) (i,j)~ez

=-~

(i,j)~oe

m(f/ Agj)logm(&/f/)- ~ m(f/ A&)logm(fi) (i,j)Eo~

k

~>-~] log m(f/)~] m(f/A &)=--~ logm(f/)m(f/A (jV=IgJ)) iefl

= -~ i~fl

j=l

i(Efl

m(f/)log m(f/) = - ~ F(m(f/)) = H,,(M). i=1

Since M ~
~=

°

Hm(~

[ ._~ ) =

/% m ( f / ) F ( m ( g j i=1 j = l

{gl . . . . .

gk} are two stochastical complete repartitio

If/)),

where

If/) t_l/

Theorem

if rn(f/) >0, if re(f/) = O.

3.2. Hm(~ v ~ I sg) = Hm(~ I ~ v ~) + H,,(~ I M) for every M, ~, ~ ~ ~.

Proof. Let M = {fl . . . . . f,}, ~ = { g l , . . . , gr}, and q~ = {hi, .. • , hs}. If re(f/hgj)>O, then

m(gj A hk If/) = m(gj A hk A f/) re(f/ A gj) = m(hk If~ A &)m(gj If/). m(f/ A &) m(f/) Moreover, for each x, y e (0, ~) we have F(x . y) = x . F(y)

+ y . F(x).

Consequently by (C6) and (C9) we obtain

Hm(~V ~ I M ) = - ~

~ ~ m(f/)F(rh(& Ahk [fi)) i=1 j = l k = l

= - ~ k ~ m(f/)F(rh(hk If/Ag~)rh(gJ If/)) i=1 j = l k = l

= - ~ ~ ~ m(f/)rh(gi ]f/)F(rh(hk If/Agj)) i=1 j = l k = l

- ~ ~ m(f/) ~ rh(hk If~ A &)F(rh(gj If,)) i=lj=l

k=l

= - ~ ~ ~ m(f/ ^ gj)F(rh(hk I f / A & ) ) - - k ~ m(f/)F(th(gj lf/)) i=1 j = l k = l

= Hm(q~ I S~ V ~) + Hm(~ I ,c~).

i=1 j = l

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Theorem 3.3. Let M, ~ e ~, M <~~. Then for each ~ ~ ~,

I ~).

Hm(S~ I c~) ~ Hm(~

(3.4)

Proof. Using the notation of the preceding theorem we have F(rh(& which along with (3.3) implies the inequality (3.4).

I h,)) = ET_-, F(rh(& ^ f [ hk)),

Theroem 3.4. Let M, ~3 e ~, M ~ ~. Then Hm(q¢ I M) ~ Hm( c¢ ] ~) for each ~ ~ ~. Proof. The function F is convex and therefore for any convex combination of elements xj ~ (0, 1), ~j o~jxj (i.e. such that o~j/>0, j = 1, 2 . . . . . Y,j o~j = 1) it holds

F ( ~ o~jxj) <~~ oljF(xj). Let ~1 = {f~ . . . . . fn},

~ = {gl .....

(3.5)

gr},

and ~ = {hi . . . . .

c~ = {i; re(f/) > 0},

fl={j;m(&)>O},

y=k;m(hk)>O},

6;={j;&~
i=1,2 .....

Put crj = rh(gj Ifi), xj = rh(hk ]g j), i, k fixed, j = 1, 2 . . . . .

jeff

h~}. Denote

r. Let i ~ o~. Then by (C9) we have

olj = ~ m(gj I f ) = ~ m(gj If) j~-fl

j=l

= ~ m(gj Afi) j=l m(fi)

m((V~=l&) ^ f / ) - - 1.

m(fi)

Since m(h A (~/j~,~,gj)) = m(h A fi) for every h c M, we get

oljxi = ~ m(gj If,-)m(hk I g~) = Y~ m(gj A f) m(hk ^gj) J~l~ J~lJ j~t3 m(fi) m(gh) = ~ m(&)m(hk A gj)_m(hk A (Vj~a'gJ) m(hk A f) - ,h(hk If,). i~a, m(fii) m(&) m(fi) m(f) Evidently EJ~t~ o~jxj= rh(hk If) also for i ~ c~. By (3.5) we obtain

FQh(hk Ifi)) <~• rh(g~ I fJF(rh(hk I&))Jet~

Therefore we have the inequality

-m(f)F(th(hk ]fi)) ~> - m ( f ) ~ th(gj

If,)F(rh(hk I&))

j=l

for i = 1. . . . .

n, k = 1. . . . .

Hm(~ I ~ ¢ ) = - ~ ]

s. Calculate:

~] m(f)F(th(hk If))

i=1 k = l

22

i=1 k = l j = l

=-~

~ m(&)F(rh(hk lgj))=H.,(~l ~).

j=l k=l

The proof is complete.

Lg,))

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Theorem 3.5. Hm( s~ [ ~:d)~ Hm( S~) for each M, ~J ~ ~. Proof. Let ~ = {1}, M = {ft . . . . . fn}. Then

m(1)F(m(f~ l l)) =

Hm(M I g ) = i=l

F(m(f~)) = Hm(M). i=1

Since any stochastical complete repartition ~ is a refinement of the repartition ~, from Theorem 3.4 it follows that H,~(~¢) = H,,(M I ~) >~Hm(M [ ~). From Theorem 3.2 and Theorem 3.4 it immediately follows:

Theorem 3.6. Hm( ~ v ~ I M) <<-Hm( ~3 [ s~) + Hm( q~ [ s~) for each M, ~, ~ • ~. Corollary 3.1. Hm(~ v ~¢) <-Hm(~) + Hm(~¢) for every ~, ~ e ~'. Proof. It suffices to substitute M = {1} in Theorem 3.6. We see that the conditional entropy of stochastical complete repartitions defined here fulfils all properties analogous to the properties of entropy of measurable repartition in the crisp case.

4. The entropy of fuzzy dynamical systems The following definition was introduced in [12, 13]:

Definition 4.1. By a fuzzy dynamical system we shall mean a quadruple (X, M, m, U), where (X, M, m) is a soft fuzzy probability space and U is a o-homomorphism (i.e. a mapping U:M---> M such that U(f') = (U(f))' and U(V~=lfn) = ~/~=l U(fn) for every f • M and any sequence {fn}S=l c M). Further U fulfils the condition

m(U(f)) = m ( f )

for e v e r y f e M.

(4.1)

The notion of a fuzzy dynamical system is non-void, since any quadruple (X, M, m, I), where (X, M, m) is any soft fuzzy probability space and I is an identity mapping on M, is a fuzzy dynamical system. Example 4.1 shows that the above notion generalizes the competent notion of classical theory.

Example 4.1. Let (X, 5e, P, T) be a dynamical system in the sense of classical probability theory. Let us consider the soft fuzzy probability space (X, M, m) from Example 2.1. If we define the mapping U:M--->M by the equality U(XA)=XT-'~A), then the quadruple (X, M, m, U) is a fuzzy dynamical system. In this case we shall say that (X, M, m, U) is induced by dynamical system (X, ~e, p, T).

Example 4.2. Let any soft fuzzy probability space (X, M, m) be given. Let T:X---~X be an m-preserving transformation (i.e. f e M implies f o T c M and m ( f o T) = m(f)). Let us define the mapping U:M--->M by U ( f ) = f o T ( U is the so-called Koopman operator). It is easy to see that (X, M, m, U) is a fuzzy dynamical system. Example 4.3. Let us consider the quadruple (X, M, m, U), where (X, M, m) is the soft fuzzy probability space from Example 2.3 and U:M-->M is defined by the equalities U(f v f ' ) = f v f ' , U(f A f ' ) = f ^ f ' , U(1)= 1, U(0) =0, U ( f ) = f ' , U ( f ' ) = f . Then (X, M, m, U) is a fuzzy dynamical system.

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357

Let any fuzzy dynamical system (X, M, m, U) be given. We define U2= U o U and by mathematical induction U " = U o U n-l, n = l , 2 , . . . , w h e r e U° is the identity mapping on M. Let U"~¢:= { U " ( f ) ; f • M} for every M • ~. It is easy to verify that U " ~ is a complete stochastical repartition. From (4.1) it immediately follows

H,,(U"M) = Hm(M),

(4.2)

H,,(M I ~)

Hm(U"M I u

for every M, N • 9.

(4.3)

Lemma 4.1. Let {a,}~=l be a sequence of nonnegative numbers such that a,+s<~ar+as for each r, s = 1, 2 . . . . . Then l i m , ~ n-~a, exists.

The proof can be found in [15]. The possibility of the definition of the entropy of a system (X, M, m, U) is based on the next lemma. Lemma 4.2. For every M • ~, l i m n ~ n - l H m ( V ~

UsM) exists.

Proof. Put a, = H,,(V;'_~~UJ~¢). According to Corollary 3.1 and (4.2) we obtain ar+s

)

j=r

~"m(;v=(l)Uj") "Jl-Hm j~r(r+S--l o'~) = at+ "m(U'(IV) UiM)) s-I ) By the preceding lemma h"m n ~ n -1 Hm(Vi=o n - - 1 UJM) exists. Definition 4.2. Let (X, M, m, U) be a fuzzy dynamical system. Then for every M • ~ we define 1

In-1

hm(U, ~ ) - - ,~limn Hm~i~ u i ~

)

The entropy hm(U) of fuzzy dynamical system (X, M, m, U) is defined by

hm(U) -- sup{bin(U, M); M • ~}.

(4.4)

One can prove the following theorem: Theorem 4.1. Let (X, 5¢, P, T) be a dynamical system in the classical sense and (X, M, m, U) be a fuzzy dynamical system induced by (X, 5¢, P, T). Then hm(U) = h(T), where h(T) is the Kolmogorov-Sinaj entropy [16, 17] of the dynamical system (X, 5¢, p, T).

Lemma 4.3. hm(U, M) = hm(U, V~-o u ~M) for every M • ~ and for every natural number k. Proof.

H ~ l j ~ . + ~ , U' h m ( U , iV=o u is6() = nlim limk+n 1 =,+~----Hm~n k + n

UiM /k+n-~ ) ,~-~, UsM =hm(U,M).

Theorem 4.2. hm(U, 5~) <~hm(U, M) + Hm(~3 [ M) for every ~l, ~ • 9.

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Proof. By means of (P3), Theorem 3.2, Theorem 3.6, Theorem 3.4 and (4.3) we obtain

V ui~ = Hm(iV° Ui~ + Hm(iVo Ui~

i =0

n'Z

i=0

nm t u i ~

oV 'u i)~ i=o

~nm(iTo u i ~

vi=O Ui~ ) "~- Z nm(Ui~[ Ui'~) i=0

nnm(~ l ~). This implies

1

n-1

)

1

n--I

t

h,,,(U, ~ ) = n....~ lim n H"(i~=o Ui~ oo ~< lim -Hm( V u i ~ .--,~ n \i=0

~- Hm(~ I ~ ) = hm(U, ~)

+Hm(~l

~).

5. The entropy of fuzzy dynamical systems and generators In this section we prove that if ~¢ • ~ is a generator of a fuzzy dynamical system (X, M, m, U), then the entropy h,,,(U) o f ( X , M, m, U) is equal to the entropy hm(U, :g) of the repartition ~1. For this we shall need some auxilliary assertions and definitions. Let any soft fuzzy probability space (X, M, m) be given. In the set M we define the relation - in the following way: for every f, geM, f - g iff m(fAg)=O, where f A g = ( f A g ' ) v ( f ' A g ) is the symmetric difference of fuzzy sets f,g. Put [f] = {g • M; m(f A g) = 0} for any f • M. It is easy to see that iffl, f2 e [f], then m ( f 0 = m(f2). In the system [M] = { [ f ] ; f • M} one can define the relation ~< as follows: for any [f], [g] e [M], [f] ~< [g] iff rn(f A g') = 0. The couple ([M], ~<) is a partially ordered set with the minimal element [0] and the maximal element [1]; moreover, [M] is a Boolean o-algebra, where [V~=lfn] is the least upper bound of a sequence {[f,]}~=l c [M], i.e. the equality V~=l [f,] = [V~=lf,] holds. Further [f] A [g] = [ f Ag]. For any [fie[M] we have If] A [f'] = If A f ' ] =[0] and [f] v [f'] = [f o f ' ] = [1] and hence [f]' = [f']. If we define the mapping ~:[M]---> (0, 1) by the equality /u([f]):= m(f) for any [f] • [M], then ju is a probability measure on the Boolean o-algebra [M], i.e. /~([1]) = 1, /~ t> 0 and [f/] ^ [fi] = [0] (i 4:j) implies #(V~=~ [f,]) = En=l /~([fn])-

Definition 5.1. A finite set [M] = {[fl] . . . . , [fnl} c [M] such that V n = l Ifi] = [1] and [f/] ^ [f~] = [0] for i :~j is called a repartition of unity of the Boolean a-algebra [M]. Since every repartition [~1] = {[fl] . . . . . [fn]} represents in the sense of classical probability theory a random experiment with the finite number of outcomes [f/] (i = 1 . . . . . n) with the probability distribution Pi = ~'£([f/l) (Pi ~ 0 and ~7=lPi= ~'=1/~([f]) =/~(V7=1 If/I) =/z([1]) = 1), the entropy of a repartition [~/] is defined by Shannon's formula

(5.1)

H.([•I) = - ~ F(t~([f/])). i=1

If [~/] = {[f~] . . . . . [f~]}, [N] = ([gl] . . . . . conditional entropy

[gk]} are two repartitions of unity of [M] we define the

H.([~t] I [~1)-- _ i=1 ~ j~= l ta([gsl)F(~([fiill[gs])),

(5.2)

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where/~([f] I [gJ])= rh(f I g J)" Let us introduce the following notation: [M] v [~] := {If//] A [gj]; / = 1 . . . . .

n, j = 1 . . . . .

k}.

It is evident that [M] v [~] = [M v ~].

Lemma 5.1. If M = { f l . . . . ,fn} is a stochastical complete repartition, then the system [ M ] = {[fl] . . . . . [f~]} is a repartition of unity of[M] and for every M, ~ • ~ we have: Hm(S~ ) = n,([6d]),

(5.3)

nm(s I a) = n.([s l I

(5.4)

Proof. For i :/:j we have m((fi Afi) A 0) = m(f/ Afj ^ 1) + m((f/ Afj)' ^ 0) = m(f/ Afj) + m(0) = O, which gives the equality [f/] m

^

[f~] = [f/, Afj] = [0] (i #:j). Moreover,

((iV=]fi)A1 ) = m ((iV=lfi)A0 ) + m ((iV=lfi)t ^ 1 ) = m ( 0 ) + l - - m (iV=lfi) = 0 ,

so that V7:1 [f,] = [VT=lf/] = [1]. This means that [M] = {Ill] . . . . . One can also immediately obtain equalities (5.3) and (5.4).

[f,]} is a repartition of unity of [M].

Definition 5.2. Let [M], [~] be two repartitions of unity of the Boolean e-algebra [M]. We shall say that [3~] is a refinement of [M] (and we shall write [M] ~< [~3]) if for every [g] e [~] there exists [f] e [M] such that [g] ~< [f].

Lemma 5.2. If M, ~ are two stochastical complete repartitions such that M <- ~ , then [M] ~< [~]. Proof. Let [g] • [~]. Then g e ~ and by the assumption there exists f • M such that g ~(0, ~) in the following way: /~(A) = lz(h(A)), A ~ b~. It is easy to verify that t~ is a probability on .9°. Since h is an epimorphism, for every [f] • [M] there is A • 5e such that h ( A ) = [f] and h(f2) A [f] = h(£'2) A h ( A ) = h(f2 A A ) = h ( A ) = [f]. Consequently h(g2) = [1]. Similarly we obtain the equality h(0) = [0]. Hence/~(12) =/~(h(g2)) =/z([1]) = 1. If An e 5e, n = 1, 2 , . . . , with A i N A j = 0 for i 4:j, then h(A~) A h(Ai) = h(Ai n A i ) = h(0) = [0] for i 4:j and therefore we have An = U h

A,

=U

h(An

=

I~(h(An)) = n=l

~(An). n=l

Lemma 5.3. Let [M] be any repartition of unity of the Boolean e-algebra [M]. Then there exists a measurable repartition cg of the measurable space (I2, be) such that h(~g) = [M]. Proof. Let [M] = {[fl], • • • , [fn]}. Since h : be--> [M] is an epimorphism, for every If/] e [M] there exists F/• 6e such that h(F~) = [f], i = 1 . . . . . n. Put

..... Go=(G,U...UG.y,

O,=G,UGo.

°-,) co=n-(p,

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Then the system ~ = {G~, (72 . . . . . G~} is a measurable repartition of the space (12, 50. We prove that h(~) = [~]. By (Cll), (C10), (C7), (C8) for k = 2, 3 . . . . . n we have m ( ( f ~ v C91 f ~ ) ) A f t ) = m ( ( f ' ~ A f ~ ) v

(191 (fi A f~)))

k-1

) k-1

< ~ m ( f ' ~ A A ) + m ( V (f, AA) ~< E m(f, A A ) = 0 i=1

and

and hence m ( ( f ~ A (i~i f 0 ) A f ~ ) = m ( ( f ~ , v (~91 f~))A f ~ ) + m(f~A (~A=~f ; ) A f~,) = 0, i . e . [f~ A

(/k~-~'f;)]

( for k = 2, 3 . . . . .

(k = 2,

= [A]

k-l)

(

3, . . . , n ) .

Calculate

k-1 ))

(k-iA:

))

[k~=if ]

n. Since

h(Go) = h((G1 U 62 U . . . UGn) ¢) =

= [1]' = [0],

we have h(G,) = h(G, U Go) = h(G,) v h(Go) = [fl] v [0] = [fd. The proof is complete. Lemma 5.4. Let M~, M2, cg ~ ~, M~ <~M2. Then there are measurable repartitions 9~l, 9~2, @ of the space (K2, 6e) such that ~2 is a refinement of ~1, h(gA0 = IS/l], h(~2 = [M2], h ( @ ) = [rg] and Hm(~ [ M~) = H~,(~ I 9~), i = 1, 2. Proof. Let M, = { f l , . . . ,fn}, M2= { h i , . . . , hs}. Since the systems [M1] = { [ f l ] , . . . , [fn]}, [M2] = {[h~] . . . . . [hs]} are repartitions of unity of [M] (see L e m m a 5.1), by the preceding lemma there exist measurable repartitions ~: = {F1, .. •, F~}, ~ = {H1 . . . . , Hs) of the space (t2, 50 such that h(F,-) = [f~], h(~.)=[hj], i = 1 . . . . . n, j = l , . . . , s . Put ~ 1 = ~ , 9 g 2 = ~ : v ~ = { F ~ f q / - / j ; i = l . . . . . n, j = 1 . . . . , s}. Then the measurable repartition ~2 is a refinement of 9~ and h(~[1) = h(.~) = [~11. h(9

2) =

v

=

v

= [M,]

v

=

v s/2] =

[ 21.

Denote by 6i = {1; h i ~
m((h~ A f ) A 0) =m(hj A f i ^ 1) +m((hj Aft)' A 0) =0, i.e. [f//] A [hA = [fj A hi] = [01. Therefore h(F~ N H i ) = ~[01 [[h~]

ifjq~61, ifje6i,

i=l .....

n.

Let ~g = {gl . . . . . gr} be any stochastical complete repartition. By means of Lemma 5.1 and Lemma 5.3 there exists a measurable repartition @ = {C1, • • •, Cr} of the space (S'2, 5~) such that h(Ci) = [g~],

D. Markechovd / Entropy of fuzzy dynamical systems and generators

i = 1,...,

361

r. Calculate

i=1 j = l

= _ ~ ~ tt(h(F~))F(ft(h(C/)[h(F~))) i=1 1=1

= - ~ ~ ~([fd)F(O([g;l I [fd))= H.([~] I [~1])= H~(~ I ~1), i=1

j=l

i=1

j=l

k=l

i=1 j = l k=l

= - ~ ~ E tKlhjl)F(fJ(tgkl)l[hd)) k=l i=1

=- ~

jEOi

~ m(h,)F(rh(gk I h D ) = H ~ ( ~ I,~2).

k=l j = l

Proposition 5.1. Let M1,

M2 . . . . .

~q~n, ~ E ~ , S~i <<-Mi+I, i = 1 , . . . , n - 1. Then there are measurable

repartitions ~I1, 9.12. . . . . 91,, @ of the space (K2, 5p) such that ~i ~< 9,1i+1, i = 1 . . . . . h(9.1,) = [M,] and Hm(~ I Mi) = n~(@ I ?1,), i = 1 , . . . , n.

n - 1, h(@) = [c~],

Proof. We prove the proposition by the mathematical induction. For n = 2 the assertion is true by the preceding lemma. Let it be valid for n = k - 1. If MI, M2 . . . . . Mk, ~ e ~ such that Mi ~< Mi+l, i = 1. . . . , k - 1, then by induction assumption there are measurable repartitions ?It, 9.12. . . . . ~k-1, of the space (g2, 5°) such that ?li ~<~1~+1, i = 1 , . . . , k - 2, h(gJi) = [M~], i = 1. . . . . k - 1, h ( ~ ) = [~¢]. By Lemma 5.3 there exists a measurable repartition ~ of the space (I2, 50 such that h ( ~ ) = [Mk]. Put ~ k = '~k--I V ~ . Then ? l k _ 1 ~ ~lk, h(gJk) = h(gJk_ 1 v ,~) = h ( ~ [ k - l ) V h ( ~ ) =

[Mk-ll v [Mkl = [Mk-i

v '~k] = [S~k]"

By means of Theorem 3.2, the induction assumption and Proposition 16.39 of [15] we obtain

/-/m(~ I ~ ) = /4"( ~e I a~-I " a~) = n,.(a~ v ~ I M~-I) - / 4 ~ ( a ~ I a~-l)

Thus the proof has been completed. ac Proposition 5.2. Let {•Q•.}.=1 be a sequence of stochastical complete repartitions such that M. <~M.+t

(n = 1, 2 . . . . ) and (The symbol

O

O

( U .oe= l [ ~ . ] )

_~-

[M]. Then for every c¢ E ~ we have lim.~= Hm(qg I M-) = 0.

( U .ze= l [M.]) denotes the Boolean o-algebra generated by the system

U~=I [M.]')

Proof. From Proposition 5.1 it follows that there exists a sequence { ,},=1 of measurable repartitions of the space (12, 50 such that ?1.+1 is a refinement of ?1, and h(?l,) = [M,], n = 1, 2 . . . . .

Denote

D. Markechovti / Entropy of fuzzy dynamical systems and generators

362

a ( g ~ = l 91,) = 5~o. Then h(5~0) =

,

= a

= o

= o

,

[Mo]

= [M].

Since h is an epimorphism, for a repartition [~¢] of unity of the Boolean o-algebra [M] there exists an 6e0-measurable repartition @ of the space (t2, 5e0) such that h ( ~ ) = [~¢]. Take into account the quadruple (~, 6eo,/~/6eo, T), where/~/6e0 is a restriction of a probability measure/~ on the o-algebra 5~o and T is the identity mapping on I2. Applying Lemma 16.46 of [15] for the dynamical system (s'-2, 5e0,/~/Se0, T), we see that l i m , ~ H~,~o(e I ~.) --0. So lim H ~ ( ~ I M , ) = lim Hta(@ ] 9g[,) = lim H~/~o(~ I 9A~) = 0. tl....~a¢

n..-.~ ~

n - - - ~ oo

The following theorem is a fuzzy generalization of the Kolmogorov-Sinaj theorem on generators (see [19]). Theorem 5.1. Let (X, M, m, U) be a fuzzy dynamical system. If M ~ ~ is a generator, i.e. M is a stochastical complete repartition such that O(Ui=o [U/M]) = [M], then hm( U ) -- hm( U , M).

Proof. Let M = { f l , . . - , fr}. Since

hm(U) = sup(hm(U, c~); qg e ~} I> hm(U, M), it suffices to prove that hm(U, ~) <~hm(U, M) for every ~¢ e ~. Put M, = VT=0 u iM. Then M, ~< Mn+l (n = 1, 2 . . . . ). Denote X = o(U~=l [M,]). Let us prove that X = o(U~=o [UiM]). Since

0 [UiMl = {[U'(fk)], k = 1 . . . . , r, i = 0 , 1, 2 . . . . } i=0

and V[,=l [ui(fk)] = [1] for i = 0, 1, 2 . . . . . it can be easily seen that ~ ~ UT=o [UiM]. Because ~ is a Boolean o-algebra, we obtain also the inclusion ~ o(UT=0 [U~M])= [M]. The reverse inclusion is evident. Thus o(U~=l [M,]) = [M]. By the preceding lemma l i m , _ . ~ H , , ( ~ J M , ) = 0 . From Theorem 4.2 follows the inequality h,,(U, ~¢) <~hm(U, M,) + H , , ( ~ I M,), n = 1, 2 . . . . . Now, by Lemma 4.3 we obtain hm(U, M,) = hm(U, M), n = 1, 2 . . . . . so that hm(U, ~) <-hm(U, M) + Hm(~ I Mn) for n = 1, 2 . . . . This implies the inequality

h,,(U, ~¢) <~h,,(U, M) + lim H , , ( ~ I Mn) = h,,(U, M), . - - . ~ oe

which ends the proof.

References [1] A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer-Verlag, Berlin, 1933). [2] K. Piasecki, Probability of fuzzy events defined as denumerable additivity measure, Fuzzy Sets and Systems 17 (1985) 272-284. [3] P. Mali~k~ and Rie~an, On the entropy of dynamical systems, Ergodic theory and related topics II (to appear). [4] B. Rie~an, On some modifications of topological entropy, Proceedings Sixth Prague Topology Symposium (1986). [5] B. Rie~an, A new approach to some notions of statistical quantum mechanics, Busefal 35 (1988) 4-6. [6] B. Rie~an and A. Dvurei~enskij, On randomness and fuzziness, Progress in Fuzzy Sets in Europe, Warszawa (1986). [7] A. Dvure~enskij and F. Chovanec, Fuzzy quantum spaces and compatibility, lnternat. J. Theoret. Physics 27 (1987) 1069-1089.

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363

[8] A. Dvure~enskij and F. K6pka, On the representation of observables in fuzzy quantum spaces, Busefal 38 (1989) 24-27. [9] A. Dvure~enskij and A. Tirp~fkov~i, A note on a sum of observables in F-quantum spaces and its properteis, Busefal 36 (1988) 132-137. [10] A. Tirp~ikovfi, The Hahn-Jordan decomposition on fuzzy quantum spaces, Busefal 31t (1989) 66-77. [11] M. Smorodinsky, Ergodic Theory, Entropy, Lecture Notes in Mathematics No. 214 (Springer-Verlag, Berlin-New York, 1971). [12] D. Markechowl, The entropy of fuzzy dynamical systems, Busefal 38 (1989) 38-41. [13] D. Markeschovfi, Isomorphism and conjugation of fuzzy dynamical systems, Busefal 38 (1989) 94-101. [14] J. Sinaj, Dynamical Systems 1. Ergodic Theory (Aarhus, 1970). [15] T. Neubrunn and B. Rie~an, Measure and Integral (Veda Bratislava, 1981) (in Slovak). [16] J. Sinaj, O poniatii entropii dinami~eskich sistem. DAN SSSR 124 (1959) 768-771 (in Russian). [17] A.N. Koimogorov, Novyj metri~.eskij invariant tranzitivnych dinami~eskich sistem, DAN SSSR 119 (1958) 861-864 (in Russian). [18] A. Sikorski, Boolean Algebras (Springer-Verlag, New York, 1964). [19] P. Waiters, Ergodic Theory - Introductory Lectures (Springer-Verlag, Berlin, 1975).