By Yuichiro Kakihara
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E. e. Since sup/(2l|2)„) is integrable by Lemma 8, we see that 7(2l|2J„) -> 7(2l|2J) in Ll, TI>1 too. (2) is derived from (1) and the fact that 77(2l|2Jn) = Jx 7(2l|2J„) a> and 77(2l|2J„) > 77(2l|2J„+1) for n > 1 by Theorem 1 (6). Now we have: Theorem 10 (Kolmogorov-Sinai). 7 / 5 is invertible and 21 G V(X) is such that V 5"2l = X, then 77(5) = H(21,5). n=—oo Proof. Let 2l„ = V 5fc2l for n > 1. Then 77(2^, 5) = #(31, 5) by Theorem 6 (2). Observe that for 25 G V(X) 77(23, 5) < 77(2l„, 5) + H^^), by Theorem 6 (4), = 77(2l,5)+77(23|2l n ) -> 77(21, 5) (n ->• oo) since H^^) I 77(23|£) = 0 by Lemma 9 (2) and Remark 2 (2).
Two algebraic dynamical systems (Tj,ipj,Uj) (j = 1,2) are said to be isomorphic if (Tj,ipj) (j = 1,2) are isomorphic algebraic measure systems via an onto isomorphism U : Ti —> r 2 such that UU\ = U2U. ,S) is a dynamical system, then (r(/i),
Let (Xi,Xi,Hi,Si) (i = 1,2) be two dynamical systems. , ip(SiXi) = S2ip(xi) for x i G Xx. In this case, (p is called an isomorphism. T h e o r e m 13. Si = 5 2 => H(Si) = H(S2). That is, the Kolmogorov-Sinai of measure preserving transformations is invariant under isomorphism. entropy Proof. Let
X2 be the isomorphism between two systems. Let 2ti G V(Xi) and observe that
(2li),S 2 ) since ^i(A) = M 2 ( V ( A ) ) for A G 2l x . Hence, H(St) < H(S2).
Abstract Methods in Information Theory by Yuichiro Kakihara