In application to all situations of the quasiperiodic motion with the rotation number given by the golden mean, the main idea of the RG analysis consists in examination of evolution operators for time intervals determined by successive Fibonacci numbers: F_k+1=F_k+F_k-1, F₀=0, F₁=1. Let us assume that in presence of noise the evolution of variable x at the GM critical point for numbers of steps F_k and F_k+1 is governed by equations

where x_n is a sequence of statistically independent random values with a zero mean and a fixed mean-square value s, e is a parameter of intensity of noise supposed to be small, y_n(x) and y_n+1(x) are some auxiliary functions. As clear, the circle map may be regarded as a particular case: for F₁=F₂=1 we set

Applying successively the maps corresponding to F_k+1 and F_k steps, we obtain for the number of steps F_k+2 a stochastic map

where the terms of the first order in the noise intensity e are taken into account.

Let us redefine the stochastic term. Let us have an orbit launched at some moment from a point x_i. Consider an ensemble of random numbers {x_i, x_i+Fk+1} with zero mean and variation s² and compose a sum with coefficient represented by functions of x_i. As the pairs {x_i, x_i+Fk+1} are statistically independent, the sum may be represented again as a random number with zero mean and variation s² multiplied by a function of x_i, namely

Let us introduce f_k+2(x)= f_k(f_k+1(x)) and rewrite the equation in the form analogous to the original one, with redefined random variable and functions f and y :

To obtain a closed system of functional equations, we square both parts of the equation and perform averaging over the ensemble of the noise realizations. As

we come to a relation

Following the main idea of the RG approach, let us perform a scale change x-->x/a^k, where a=-1.288574553... is a known universal orbital scaling factor for the GM critical point. Then, in terms of the redefined functions

the above equations imply

These relations define the RG transformation for the set of functions {g_k, f_k, Y_k, F_k}. The routine may be repeated again and again to obtain the functions for larger and larger k, i.e. to determine the renormalized evolution operators for time intervals corresponding to iteration numbers F_k.

As known, at the critical point GM the functional sequence g_k, f_k converges to a fixed point of the RG transformation determined by the equations

or

Convergence of the functions g_k, f_k to the fixed-point solution implies that the recursive linear equations for the functional pairs {Y_k, F_k} asymptotically will correspond to an eigenvector associated with the largest eigenvalue W for the next eigenproblem

A numerical solution yields W=5.31849047771... Now, in linear approximation in respect to the noise amplitude the stochastic map corresponding to F_k and F_k+1 steps of iterations at the critical point GM may be represented in the renormalized variables as

where g=W^1/2=2.30618526526. Taking into account additional perturbations of the evolution operator caused by parameter shifts from the critical point, one can derive from this the scaling law formulation given in the main text.