By R. L. Dobrushin, Y. G. Sinai
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Abdulla-Zadeh, R. A. Minios, S. K. 49) be a Banach space of functions on Í (Z^ with the norm [¢11 = sup ^ where g is some constant I I I¢(1) I diam I (2. 50) By using the estimate (2. 48) we obtain that for sufficiently small ß the operator D in Í Y, (D ^ d )= r< i (2. 5 I) Ф(1’) ^ V-I has norm <1 and hence, there exists an inverse (E + D) = E + G. V j. Л where G^, is a matrix of the operator G. (2. 52) Hence IlvJ (2. 53) where K is some constant, the series 2)^i' ^i< (^°^ abritrary I) converges with respect to the norm of L_(C(Z^), ir ) for properly C.
T C Z I with respect to the Gibbs measure P^ satisfy ^ t' ß,p. 59) , B are constants and r(ß) is some function such that r(ß) — 0 when ß -^ 0 . Before proving this lemma, we shall show how it implies the Lemma 2. 4. I 2 Indeed, note that for some к C Z and for any I C (Z ). p(I) =0 24 F. H. Abdulla-Zadeh, R. Л* Minios, S« K. Pogosian and hence, as we may see from (I. 15), o)(I) = O I C (Z ), for some 1 ^ к e Z . Thus, using (I. 14) and (2. n (z \ ^ 4 jï( i= Jc= 1,2,... I 4 ß- Put b ( I ,r ) = 0)(1) (2.
C or C T and P be the measure on (X ), The meaa ’ O' sure P* defined on (X , ) is said to be a projection of the meaA A sure P if P '(r) = P ( { x : x 6IX , X e Г}), Qf A re^ A If P^ and P^ are the probability measures on the spaces 42 V. Ya. Basis (X , ¾ ' ) and a ’ (X P P respectively and X Cor f lß let ■“ I S PX y)P(dx, dy) \ \ where the greatest lower bound is computed over all the measures P on (Х ^ Д ^ )Х (X ^ ,^ ^ ), suchthat Р ( Г х Х ^ ) = PJ(F), P(X^ X Г) = P^(F), F e ^ * ^ , where PJ and P^ are the projections of the mea sures P^ and P^ onto (X ^ ,^ ^ ).