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29) in the even subspace. 30) -N according to the principles we have mtated in the case of the free energy. )) lim N -~ ~ log I~N - 0critl + log MLI,6N•aI~IXN6NhN + N . 32) log k 2 if log M~I•~N,al@lkN6NhN+ a2e2k26 N as N ~ ~ . s. 31) hold. Indeed, log a~ 1, 86 1N ~, _ _ + a2~2 r~ ~ . But 0 if D ~ O, as the expectation of S in an odd measure, and it is bounded by continuity. 33) remains true if N ll BN hN = exp(-g(N)) with g(N) > O, g(N) ~ 0 as N ~ N Assume now, as befere, M~,h,f = that the thermodynamic N lira -~ o~ MN, ~,h, f limit 56 exists.

E. its critical indices. These are then the results of Method i . It is only in Method 2 that another fixed point (the Selutlon 2) will play a role, and that the "crossover" (the flow i under J~from the fixed point ~ to the fixed point (4w/c)W6) has to be studied. This will yield the proof of the existence of the thermodynamic limit and the presence of a phase transition. All the above considerations have a straightforward tion to points c = 2 I/j, j = 2,3, . . poly~omial , and again a fixed point theorem applies.

Math. Phys. (1975). G. H. RABINOWITZ : Bifurcation from Simple Eigen- values. J. Funct. Anal. 8 , 321 (1971). The proof of Bleher and SinaZ used the "method of the separatrix". Improving slightly on t h e i r m e t h 0 d [18] , we showed in P. -P. ECKMANN: The ~-Expanslon for the Hierarchical Model. Commun. Math. Phys. 55, 67 (1967). that ~c is a C N function of c > 0, for all N and s sufficiently small, so that the c-expansion for the critical indices, and more knowledge about ~c follows. The proof of the existence of ~c we give in these Lectures Notes is new and has not appeared before.

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