Existence of analytic invariant curves for a complex planar mapping near resonance

In this paper a 2-dimensional mapping is investigated in the complex field C for the existence of analytic invariant curves. Employing the method of majorant series, we need to discuss the eigenvalue alpha of the mapping at a fixed point. Besides the hyperbolic case vertical bar alpha vertical bar not equal 1, we focus on those alpha on the unit circle S-1, i.e., vertical bar alpha vertical bar = 1. We discuss not only those alpha at resonance, i.e., at a root of the unity, but also those alpha near resonance under the Brjuno condition.