Local dynamics of parabolic skew-products

TL;DR

This paper investigates the local dynamics of parabolic skew-product maps in complex variables, providing formulas for conjugacy maps near the fixed point, bridging the understanding between one-dimensional and higher-dimensional complex dynamics.

Contribution

It introduces a detailed analysis of parabolic skew-product maps, offering new formulas for conjugacy maps that describe orbit behavior near the fixed point.

Findings

Formulas for conjugacy maps in different regions

Description of orbit behavior around the origin

Bridging gap between one and several complex variables

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.