Local dynamics of parabolic skew-products

Abstract

The local dynamics around a fixed point has been extensively studied for germs of one and several complex variables. In one dimension, there exist a complete picture of the trajectory of the orbits on a whole neighborhood of the fixed point. In dimensions larger or equal than two some partial results are known. In this article we analyze a case that lies in the boundary between one and several complex variables. We consider skew product maps of the form F (z, w) = ({\lambda}(z), f (z, w)). We deal with the case of parabolic skew product maps, that is when DF(0,0) = Id. Our goal is to describe the behavior of orbits around a whole neighborhood of the origin. We establish formulas for conjugacy maps in different regions of a neighborhood of the origin.