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Supplement and correction to the preceding paper ``A functional method for stationary channels''

1964; Tokyo Institute of Technology; Volume: 16; Issue: 3 Linguagem: Inglês

10.2996/kmj/1138844920

1881-5472

Hisaharu Umegaki,

DNA and Biological Computing

In the second part of Theorem 6 of the paper in the title, it was stated that, under (C4), the capacity C s can be achieved on P(X, S).In the proof of this, we shaw that H(p), H\p) and H"{fi) are weakly* upper-semicontinuous on P(X, S), by which we have conckided impatiently that the weak* upper-semicontinuity of R(p).The proof of this conclusion is incomplete, and hence the achieving of C s on P(X, S) is yet unkown for such channel distribution.Therefore, in Theorem 6, the statement relative to the part for (C4) should be putout.In order to study the theorem for the imput X and the output Y being general discrete dynamical systems which have not clopen bases or more generally clopen partitions, we shall give the following theorem which corresponds to the first half of Theorem 6. THEOREM.// both the input X and the output Y are compact metric spaces with the homeomorphisms S and T respectively, in which the channel distribution v(-, •) is defined by (Cl), ( C2) and (C3).Then, for every finite Borel measurable partitions'2 of X and 2 of Y, the transmission functional 9t( ; 2, 2) is defined and the equality (19) (G(2, 2)=sup{*R(£; 2, g); peP e (X)}) holds.Now, we remark that the entropy functionals H{-, 3, S) and H{ , 2, T) are defined and so is 3t( ; 2, 2) for every 2 and 2 which are not necessarily clopen, and also remark that theorems (Theorems 2, 4 and the first part for (C / 1) of Theorem 6) given in the paper mentioned in the title (and also Theorem 5 in the paper [3]) are available for the present (X, 2, S) and (Y, 2, T).Under these considerations, the theorem in this paper is proved, along the method of Parthasarathy [2], by the similar way of Theorem 6 for (Cl).Indeed, denote R (cl) the set of all transitive points reX in the sense of [1] (this is the case that R is the set of all regular points rzX relative to C(X) in the notation of [2] or [3]) which satisfies p(R) = l for every peP(X, S).Then for any bounded Borel measurable function f(x) on X x)Λ \ f(x)dm r (x)dp(r) JRJX