(06/24/2011, 04:36 PM)sheldonison Wrote: Update, one Thesis paper I started to read, written by Edgar Arturo Saenz Maldonado on the Brjuno number seems to have the formulas.
\( \lambda=\exp(2\pi i \alpha) \)
\( f(z)=\lambda z + \sum_{n>=2}a_n z^n \).
And .... the formal power series of h {the Seigel disc function} is given by
\( h(z)=\sum_{i>=1}h_i z^n \)
If h is the solution of the functional equation ... \( f(h(z))=h(\lambda z) \), the coefficients of the series must satisfy (formally) the following recursive relation:
\( h_n \)=1, for n=1, and for n>=2,
\( h_n = \frac{1}{\lambda^n-\lambda}\sum_{n=2}^{n}a_m \sum_{n1+...+n_m=n} h_{n1}h_{n2}...h_{n_m} \)
where in the second summation, \( n_i>=1 \)
"... By the formulas in question, it is possible to determine the coefficients of the formal power series of \( h_f \); the denominators of these coefficients can be written as products of the form \( \lambda^n-\lambda \), for n>=2, since \( \alpha \) is an irrational number these products could be very small....", which is where the Brjuno number comes from.
The function h is the (inverse of the) Schöder function of f. Its well-known that the case of multiplier \( f'(z_0)=e^{2\pi i\alpha} \), \( \alpha \) irrational, behaves similar to hyperbolic fixpoints (i.e. \( |f'(z_0)|\neq 0,1 \)).
