By Pierre Collet

ISBN-10: 3540086706

ISBN-13: 9783540086703

ISBN-10: 3540358994

ISBN-13: 9783540358992

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22) Using the quadratic nature of J~c' and the definition of ~2J~(c(c),~) ~ = 2 J~c(~,~' ) cf. 13) , we set ~$,e = ~ = ~2J~{c(~),~). 6. For all N > 0, there is an ~o(N) > 0 such that for 0 < e < ~o(N) the operator (~f with norm less than C N C 12. 22) posses- ses thus a unique fixed point. We have thus shown the existence of ~ . ) + O(e (N-i)/2) in L . Note that this is only existence for c > 0, so we do not have a bifurcation into two solutions as would be implied by perturbation theory alone.

12) most elegantly leads to a solution of the problem in perturbation theory, not even the most advanced versions of the implicit function theorem seem to be sufficient for our problem. 17). On the other hand, whenever seem to have elgenvalues. J~ is bounded, 92~ Put in another way, neighborhoods does not are too big in function spaces. One could also say that the polynomial approximation @(~)~ I + a 04 does not push us sufficiently into the direction of the bifurcating branch for the implicit function thm.

Lecture Notes in Mathematics, Vol. 583, Berlin#Heidelberg, New York. Springer Verlag (1977). The construction of the normal form is discussed in [ 26] E. NELSON : Topics in dynamics, I. Flows , Mathematical notes, Princeton. Princeton University Press (1969). J. W E G N E R : Corrections to scaling laws, Phys. Rev. %) of Sternberg [26]. 5. Discussion of the Critical Indices One of the triumphs in the RG approach has been the correct prediction of experimentally measured critical indices. The critical indices are defined as follows : Let Q(6) be some physical quantity de- pending on the inverse temperature 6 = I/kT, where k is the Boltzmann constant.

### A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics by Pierre Collet

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