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renormalization group method
golden-mean RG analysis
Let us consider a one-dimensional map 
.
The function 
determines evolution operator for one step of discrete time, it is supposed
to have a maximum at the origin. For two steps we obtain 
,
and let us introduce instead of x a new variable rescaled with factor
. Changing
x to
in both sides of the equation, we write 
,
where 
.
Now let us take g1(x) as a new initial function and
produce the same operations. Then, we obtain a renormalized evolution operator
for four steps: 
,
where 
.
Repetition of the procedure yields a recurrent functional equation
.
If the original map g0(x) depends on parameter and
demonstrates period-doubling cascade, then, at the accumulation point of the
period-doubling, with appropriate selection of rescaling constant a we have
. The limit
function g(x) will satisfy to the Feigenbaum-Cvitanović
equation
,
i.e. it is a fixed point of the functional equation. If the maximum at the origin is quadratic, it will correspond to the Feigenbaum criticality (type F); powers 4, 6, 8 correspond to other universality classes (T, S, E).
Let us now search for solution of the functional equation close to the fixed
point in a form 
.
At 
we have
.
This equation has structure of linear operator equation 
.
If we have some eigenfunction h(x) and eigenvalue n satisfying
, then a
solution may contain a component that behaves in a course of iterations as 
.
Those eigenvalues will be of relevance, which are larger in absolute value than
1 and are not associated with infinitesimal variable changes. A number of relevant
eigenvalues corresponds to codimension of the given type of criticality.
The eigenvalues themselves define parameter space scaling factors in
some special local coordinate system (scaling coordinats).
Generalization of the above analysis for two-dimensional mappings 
,
consists
in the following. We select appropriate coordinates on the plane (x,
y) and construct a functional sequence
If it converges (with properly selected
and 
) to
a fixed point, or to a periodic orbit, it will correspond to some criticality
type and universality class. Examples are types В,
BT, H, FQ,
C from our collection.
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