By Lars V. Ahlfors

ISBN-10: 0070006563

ISBN-13: 9780070006560

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**Example text**

W. ]. Important work on the representation of endomorphisms as factors of Bernoulli endomorphisms appears in [ R']. The information function IS = I(0 1S 1(B) of an endomorphism S of (X, 03, m) is clearly an isomorphism invariant which can be used directly, without the complications of an additional coboundary. Indeed, many endomorphisms are completely characterised by the multivariate distributions of IS, is o S, is o S2, ... (see [P. ]). 37 CHAPTER III: FINITARY ISOMORPHISMS 1. THE MARKER METHOD ([K.

XU to V1 X ,,, X Vi by setting i (S1 x... ) = Si(b1) X ... ) J J 4. J is a society. it x * S1 X ... X Si is a society. Also, (S1 X , . ) = S1 X , , , XS Proposition [K. 1 1. (ii) (i) S j (iii) For any society S from U to V there is a society R < S such that card { g E V : Hb1, b2 E U, b1 * b2, with g E R(b1) n R(b2) I < card U. S Proof. (i) Let G C V, then b r{ S (G) iff G n S(b) _ 0, so that * (G)C= { b E U : S(b) C GC }. Hence P(S(G)C) =1-P(S(G)) =P{b:S(b) C GCJ (Y (GC) =1-Q (G) P(S (G)). The second part of (ii) is straightforward.

Then B1 C Cn whose future length is less than (BI) and B1 E " T1-1 a. We have f N on -N 00 U B' and f >N on its complement, which must therefore be A' Hence N+1 1=1 l B1' U A1, B2', Bg', ... 46 (iv) , " T1-1 a) 5 2 Z aN+i+l < 2 1 ak . 00 j=-N i=0 00 n T1-0 1[, d(" k=N+l i=0 The result follows by letting n '' 00 and using 2. 46( vi) . // 19. Theorem [P. 7]. , 03,, m. ) (i = 1, 2) whose state partitions have 1 1 1 1 finite entropies. If 0 has finite expected future and inverse future code-lengths then the processes are quasi-regularly isomorphic through 0.

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