The Existence of Analytic Solutions of a Functional Equation for Invariant Curves

摘要

This paper is concerned with an iterative functional equation from the problem of invariant curves. By reducing the equation with the Schroder transformation to another functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant λ given in the Schroder transformation is required to fulfil that λ is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we obtain analytic solutions in the case of λ at resonance, i.e., at a root of the unity and the case of ~λ near resonance under the Brjuno condition.