Lindstedt Series and Kolmogorov Theorem

the KAM theorem from a combinatorial viewpoint. Lindstedt, Newcomb and Poincaré introduced a remarkable trigonometric series motivated by the analysis of the three body problem, [P]. In modern language, [G2], it is the generating function, that I call Lindstedt series here, ~h(~ ψ) = ∑∞ k=1 ε k~h(k)(~ ψ) of the sequence ~h(k)(~ ψ) of trigonometric polynomials associated with well known combinatorial objects, namely rooted trees. It is necessary to recall, first, the notion of rooted tree as used here. We lay down one after the other on a plane k pairwise distinct unit segments oriented from one endpoint to the other (respectively the initial point and the endpoint of the oriented segment also called arrow or branch). The rule is that after laying down the first segment, the root branch, with the endpoint at the origin and otherwise arbitrarily, the others are laid down one after the other by attaching an endpoint of a new branch to an initial point of an old one and leaving free the branch initial point. The set of initial points of the object thus constructed will be called the set of the tree nodes. A tree is therefore a partially ordered set with top point the endpoint of the root branch, also called the root (which is not a node). The angles at which the segments are attached will be irrelevant: i.e. the operation of changing the angles between arrows emerging from the same node (each arrow carrying along, unchanged, the subtree of arrows possibly attached to its initial point) generates a group of transformations and two trees that can be overlapped by acting on them with a group element are regarded as identical. The number of trees with k branches is thus bounded by 4k!. With each tree node v we associate an incoming momentum, or ”decoration”, which is simply an integer component vector ~νv; with the root of the tree (which is not regarded as a node) we associate a label j = 1, . . . , l. With each branch λ = vv, with final point v and initial point v, we associate another integer component vector, the branch momentum ”flowing through the branch”, defined by ~ν(v) = ∑ w≤v ~νw (we shall also denote ~ν(v) by the symbol ~ν(λ)). Then, given a positive matrix J and a trigonometric polynomial f(~ ψ) = ∑ 0<|~ν|<N f~ν cos~ν · ~ ψ, f~ν = f−~ν, we consider from now on only decorated trees θ with k branches, such that ~ν(v) 6= ~0 for all v, and associate with each decorated tree the value: