In this paper, we study the continuity of rational functions realized by Büchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function f has at least one point of continuity and that its continuity set C(f ) cannot be computed.In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational Π 0 2 -subset of Σ ω for some alphabet Σ is the continuity set C(f ) of an ω-rational synchronous function f defined on Σ ω .

In order to push the study of solutions of nonlinear differential equations involved in quantum electrodynamics 1 , we focus here on combinatorial aspects of their renormalization at {0, 1, +∞}.

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer n ≥ 1, there is a D ω n (Σ 1 1 )-complete tree language L n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Büchi tree automaton must be Borel. Then we consider the game tree languages W (ι,κ) , for Mostowski-Rabin indices (ι, κ). We prove that the D ω n (Σ 1 1 )-complete tree languages L n are Wadge reducible to the game tree language W (ι,κ) for κ − ι ≥ 2. In particular these languages W (ι,κ) are not in any class D α (Σ 1 1 ) for α < ω ω .

Abstract. Let X be a separable Banach space and denote by L(X) (resp. K(C)) the set of all bounded linear operators on X (resp. the set of all compact subsets of C). We show that the maps from L(X) into K(C) which assign to each element of L(X) its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where L(X) (resp. K(C)) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us to derive the topological complexity of some subsets of L(X) and to discuss the properties of a class of strongly continuous semigroups. We close the paper by giving a characterization of strongly continuous semigroups on hereditarily indecomposable Banach spaces.