Small divisors problem in dynamical systems and analytic invariant curves for an iterative equation related to a rational difference equation (Q904338)

The existence of analytic invariant curves of the iterative equation \[ f(f(x)) = \frac {ax}{x f(x) +b}, \quad a, b > 0, \] is studied. Using the so-called Schröder transformation \(f (x) = g (\alpha g ^{-1} (x))\), the iterative equation is reduced to the auxiliary equation \[ g (\alpha^2 x) (g (\alpha x) g (x) + b) = ag (x). \] Next, the existence of analytic solutions in a neighborhood of the origin of the auxiliary equation is obtained under one of the following conditions: {\parindent=8 mm \begin{itemize} \item[(C1)] \(0 < | \alpha| < 1\); \item [(C2)] \(\alpha = e ^{2 \pi i \theta}\), \(\theta \in \mathbb R\backslash \mathbb Q\), and \(\alpha\) satisfies the Bryuno condition: \[ \sum_{\nu \geq 0} q_\nu^{-1} \log \omega (q_{\nu + 1})^{-1} < \infty, \] where \[ \omega (m) = \min_{2 \leq n \leq m} \{| 1 + \alpha^n|\}, \quad m \geq 2, \] and \(\{q_\nu\}^\infty_{\nu = 0}\) is a sequence of integers with \( 1 = q_0 < q_1 < \dots \); \item [(C3)] \(\alpha = e^{2 \pi i q/ p}\) for some integer \(p \in \mathbb N\) with \(p \geq 2\) and \(q \in \mathbb Z \backslash \{0\}\), and \(\alpha \neq e^{2 \pi i \xi /v}\) for all \( 1 \leq v \leq p - 1\) and \(\xi \in \mathbb Z \backslash \{0\}\). \end{itemize}}