The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces

We consider r-ramification ultradifferentiable classes, introduced by J. Schmets and M. Valdivia in order to study the surjectivity of the Borel map, and later on also exploited by the authors in the ultraholomorphic context. We characterize quasianalyticity in such classes, extend the results of Schmets and Valdivia about the image of the Borel map in a mixed ultradifferentiable setting, and obtain a version of the Whitney extension theorem in this framework.

A PROJECTIVE DESCRIPTION OF THE SPACE OF HOLOMORPHIC GERMS

A natural topology on the set of germs of holomorphic functions on a compact subset $K$ of a Fréchet space is the locally convex inductive limit topology of the spaces $\mathcal{O}(\sOm)$ endowed with the compact open topology; here $\sOm$ is any open subset containing $K$. Mujica gave a description of this space as the inductive limit of a suitable sequence of compact subsets. He used a set of intricate semi-norms for this. We give a projective characterization of this space, using simpler semi-norms, whose form is similar to the one used in the Whitney Extension Theorem for $C_\infty$ functions. They are quite natural in a framework where extensions are involved. We also give a simple proof that this topology is strictly stronger than the topology of the projective limit of the non-quasi-analytic spaces.AMS 2000 Mathematics subject classification: Primary 46A13; 46F15

Regularity of invariant graphs for forced systems

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is a special case of the well known concept of an inertial manifold for more general systems. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. The aim of this paper is to extend this result in two directions: firstly, where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney extension theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.