By Pierre Collet
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Additional resources for A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics
5. 4) and hence the flow defined by ~f = %( ~ + ^ f) - ~ can be llnearized on the unstable even and odd subspace and one has A AA -i A ^ ~(~)A ^ Es See the note on page 135. W. C. PUGH, M. SHUB : Invariant Manifolds. 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.
Due to the particular of the model, definition we have for J < N < SoS2J > 2 N f c < : < (So+Sl)/2 ' (s2J + s2J+1)/2 >N 2,f So s2J-I > 2N-I' JT~ S)(f) In the scaling limit, < as j ~ ~ . This index we fix N' = N-J : SoS2J > 2N'+J f ' ~ = 8crit " and write c-J < So 2 > 2 N', ~c~)J(f) . Then we get log < SoS2J > 2N'+J f lim = J ~ ~ _ log2c log 2 j s i n c e ~ p ~) (f) converges to the image of ~c By definition log lim J ~ ~ Remarks = " 2 - d - ~ " , so that "~" = i + log2c log J on Section 5: Our treatment part, < SoS j > an expansion is a precise version of standard of the discussion in  arguments, and, in .
2. 3) CE s X is small in norm as one gets near the fixed point. We can visualize the statement of the theorem as follows. ) corresponding to the eigenvalues of ~)T=~)J~(e~) which are smaller and bigger than one, respectively. 40 Eu \ ~ Fig. 5 l = "relevant directions" u Es ~s = "critical surface" The transformation S maps %Ds onto Es, S T S -I is equal to the linear map ~ J ~ ( ~ ) ~u onto E u , and then plus a remainder which leaves Es, E u invarlant and is small. All these statements hold in a neighborhood of radius 0 ( c 90) of the origin of L We would like to eliminate now the non-linear (small) remainder by a further coordinate transformation.
A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics by Pierre Collet