To start the RG analysis, let us consider a map

where

Let us apply this map twice and assume that the noise intensity parameter k is small. In the first order in this parameter we obtain:

We may interpret the expression in square brackets as a new random variable and designate it as U₁(x)x'_n. We select the factor U₁(x) in such way that the value x'_n to have the same variance s as x_n.

Because of statistical independence of x_n and x_n+1, the mean squares of the two terms of the summ are simply added. As follows,

Thus, we reduce the equation for the two-fold iteration to the same form as the original one,

but with new functions

The performed procedure may be applied many times. It yields a sequence of functions g_k, U_k that satisfies a chain of recurrent functional equations

In accordance with the Feigenbaum theory, the sequence g_k converges to a limit function g, which represents a fixed point of the functional equation of Feigenbaum and Cvitanović

In the second equation at large k, we may substitute g instead of g_k. Searching the solution in a form

we come to the eigenvalue problem

It may be solved numerically as the function g and the constant a are known. The largest eigenvalue is g=6.619036513, and the respective eigenfunction is plotted in the diagram.