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

We study the existence of analytic invariant curves of the iterative equation f(f(x)) = ax/(f(x)x + b), which arises from second-order rational difference equations. By reducing the equation with the Schröder transformation to an auxiliary equation we discuss not only the constant α at resonance, that is, at a root of the unity, but also those α near resonance under the Bryuno condition.