Smoothness of invariant manifolds

Abstract

In this paper we present some functional analytic tools that allow us to prove a theorem on C k,η -smoothness of general (discrete and continuous) invariant manifolds for C k,η -systems. A crucial role in this proof will be played by a theorem on C k,η -parameter dependence of a fixed point of a contraction defined on a scale of Banach spaces. This fixed point principle is stated independently and may be of interest in itself. Further functional analytic considerations concerning smoothness properties of the Nemytski operator are presented in a rather general form. Throughout the paper we treat the two cases of continuous and discrete dynamical systems in a unified manner. Thereby we use some concepts and results from the so-called calculus on measure chains which was developed by the author in Res. in Math. 18, 1990, 18–56. Knowledge of this calculus is not necessary for understanding the paper.