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 Infinite tetration of the imaginary unit bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 06/26/2011, 08:06 AM (This post was last modified: 06/26/2011, 08:06 AM by bo198214.) (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. $|f'(z_0)|\neq 0,1$). « Next Oldest | Next Newest »

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