Analytic Invariant Curves in a Two Dimensional Complex Discrete Dynamical System

In this paper, we are concerned with the existence of analytic invariant curves of a 2-D mapping in the complex field C. Employing the method of majorant series, we need to discuss the eigenvalue alpha of the mapping at a fixed point. We discuss not only these alpha at the hyperbolic case 0<vertical bar alpha vertical bar<1 and resonance, i.e., at a root of the unity, but also those alpha near resonance (i.e., near a root of the unity) under Brjuno condition.