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Non-accumulation des cycles-limites. I. II. (Nonaccumulation of limit- cycles. I. II.). (French) Zbl 0615.58011
These two notes offer a proof of a ”conjecture” put forward by H. Dulac way back in 1923 and saying that the limit cycles (or isolated cycles) of a real analytic vector field can never accumulate. Since accumulation could occur only near a polycycle or ”separatrix” (i.e. a piecewise analytic trajectory of the field, which may reduce to a single critical point), this amounts to saying that the return map \(x\to f(x)\) associated with a polycycle is either \(\equiv\) x or without fixed point on a small interval ]0,\(\in [\). Actually, Dulac stated this result as a theorem, but the proof he gave in support of his claim doesn’t stand up to scrutiny.
Our own proof shuttles back and forth between two fundamental objects the return map f, which is an honest point mapping, and its formal counterpart \(\tilde f,\) which carries not just the asymptotic expansion of f, but also ”inaccissible” or ”transasymptotic” terms. We show that f and \(\tilde f\) can be reduced to a product of basic factors which belong to certain quasianalytic groups. Quasianalyticity, in turn, ensures that f cannot oscillate near 0. Hence the finiteness of its isolated fixed points. Central to the argument is a formal lemma on the factorization of mixed power series (made up of exponentials, logarithms, and ordinary powers) and two further independence lemmas, which rest on the theory of resurgent functions.
The detailed proofs, along with numerous additions, will appear in a forthcoming paper entitled ”Le problème de Dulac: solution et retombées”. One noteworthy by-product of this study is the notion of compensation which, we can’t help feeling, far exceeds Dulac’s problem both in scope and importance. The idea, roughly speaking, is that whenever, in a local analytic situation, the search for a well-defined geometric object leads to divergent series (owing to small denominators or other causes), then divergence can always be overcome through a skilful regrouping of terms and/or proper resummation methods. To work out this program, we introduce and study the so-called compensators. Then we apply them to fairly general problems, such as constructing the invariant varieties of local vector fields or diffeomorphisms, in any dimension. We also think that they might play an important part in the solution of Hilbert’s 16th problem (second part).

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems