Local dynamics of parabolic skew-products

Abstract

The local dynamics around a ?xed point has been extensively studied for germs of one and several complex variables. In dimension one, there exist a complete picture of the trajectory of the orbits on a full neighbourhood of the ?xed point. In greater dimensions some partial results are known. In this paper we analyze a case that lies between one and several variables. We consider skew product maps of the form F (z, w)=( (z),f(z, w)) and deal with the parabolic case, that is, when DF (0, 0) = Id. We describe the behaviour of orbits around a neighbourhood of the origin. We establish formulas for conjugacy maps in diferent regions of these neighbourhoods.