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The proof may also be found in standard textbooks such as Brown [3], Ornstein [3] and Walters [1]. Example 16 (Markov shifts). Consider a finite scheme (Xo,p) and the inifinite product space X = XQ with the shift S as in Example 14. , my > 0, Yl mij = 1 for 1 < i, j < £, and m = (mi,... j=l , mi) t be a probability distribution such that Yl mifnij = rrij for 1 < j < £. For each i,j, i=l rriij indicates the transition probability from the state dj to the state aj and the row vector m is fixed by M in the sense that m M = m.
Theorem 6. The entropy functional H(-, 21, S) is a nonnegative bounded linear functional on Ma(X). Remark 7. Here are some properties of the entropy functional. (3E:). Then: (1) « C » « , => H{t, 2t, 5) < H{t, 53, S). (2) 21 < 53 => ff(£, 21,5) < #(£, 53,5). i3) Hit,S-1K,S) = Hit,VL,S). (4) # ( £ , a v 9 3 , S ) < . ff(£,2lV«B,S). The entropy of 5 under t £ -Mi1" (X) is defined by Hit, s) = sup {#(£, a, 5): a e P(x)}. If a e ViX) is such that 2 ^ = X, then H(S,S)=H{t,X,S), f£M+(X) by Remark 7(1).
V^S-'a) < 77(2l| V 5 - i 2 i ) < 77(2l|2) = 77(21) 22 Chapter I: Entropy forn > 1 and hence lim H (oil V 5 _ J '2l) exists. 7=0 / = lim - YH(*\ V S-'a) = lim fffal v s^'a). The second equality is obtained from Lemma 4 (2). (2) An application of Theorem 1 (11) and (1) above give H(21, s)= lim ifffVs-'o) n-t-oo n \ j=0 n->oo n 1 V /n-1 I = lim ±i7(V("-1>"vV