##
**Six lectures on transseries, analysable functions and the constructive proof of Dulac’s conjecture.**
*(English)*
Zbl 0814.32008

Schlomiuk, Dana (ed.), Bifurcations and periodic orbits of vector fields. Proceedings of the NATO Advanced Study Institute and Séminaire de Mathématiques Supérieures, Montréal, Canada, July 13-24, 1992. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 408, 75-184 (1993).

New original so-called resummation methods are described. These methods have numerous applications to the theory of differential and functional equations, differential geometry and local dynamical systems. Some of them are closely related with a proof of the famous Dulac’s conjecture about the finiteness of limit cycles.

One can divide this manuscript into two parts. The first part consists of three lectures where basic notations, definitions and results from previous author’s works are described [see the author, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques, Hermann, Paris. (1992; Zbl 1241.34003)] and [Compensation of small denominators and ramified linearization of local objects’, Astérisque V. 222, 135–199 (1994; Zbl 0810.58036)]. In particular the notions of resurgent functions, alien calculus, medianization, compensation and seriation, moulds and comoulds, arborification are discussed. Then the three main facts concerning local objects, in other words, germs of vector fields or diffeomorphisms of \(\mathbb{C}^ m\) are recalled. Thus, it is well known that local objects are formally linearized in the non-resonance case. Moreover if such an object isn’t quasiresonance then it can be analytically linearized. In the quasiresonance case the convergence of the formal linearizators may be restorted by means of the arborification. Any nondegenerate local object can be linearized by means of a seriable- ramified change of coordinates, that is, by substitutions \(y_ i = x_ i(1 + \varphi_ i (x))\) which are formal series in variables \(x_ i, \log x_ i\). Above all it is also supposed that such a series must be summable in a unique way, by means of suitable Borel-Laplace procedure, to a sum that is defined and analytic in a ramified slowly spiraling neighborhood of the origin in \(\mathbb{C}_ \bullet^ m\) where \(\mathbb{C}_ \bullet\) is the Riemann surface of the logarithm.

The second part also consists of three lectures. A construction of the fields of transseries and analysable germs is described in detail. Transseries are series of transmonomials, that is, elements of the scale of infinity in the sense of Bourbaki. The basis of the scale contains \(z^ \sigma\), \(\sigma \in \mathbb{R}\), and it is closed under the log- and exp-operations. The algebra of transseries \(\mathbb{R}[[[z]]]\) may be characterized as being the smallest algebra that i) contains \(z\) and \(\mathbb{R}\); ii) is stable under the precomposition by exp-operation and the postcomposition by log-operation; iii) contains \(\mathbb{R} [[\varepsilon_ 1, \dots, \varepsilon_ r]]\) as soon as it contains transseries \(\varepsilon_ 1, \dots, \varepsilon_ r\) (for \(\varepsilon_ i (\infty) = 0\), formally). The next challenge would be to prove that formal solutions of any analytic dynamical system have canonical analysable expansions. A ramified analytic germ having asymptotic transseries is called the analysable function. The procedure of summation of transseries into an analysable germ is known as the accelero-synthesis, and reverse procedure is known as the decelero-analysis. This procedure consists in iterative executing of the following five steps: 1) changing of independent variables to sufficiently low ones; 2) carrying out the formal Borel transform; 3) changing of the Borel variable to the corresponding fast variables with the aid of an integral transformation; 4) performing the Laplace transformation; 5) reverting to the original variable. The summation is a constructive correspondence which is compatible with the canonical embedding. A proposed simple and conceptual proof of non-accumulation of limit cycles of real-analytic, first-order differential equations is based on the last properties. The return map turns out to be analysable and hence it is either identical or monotonic as well as its transseries.

It should be remarked that a similar theory of transseries called compound asymptotics has been developed in an earlier work [A. N. Kuznetsov, Funct. Anal. Appl. 23, No. 4, 308–314 (1989); translation from Funkts. Anal. Prilozh. 23, No. 4, 63-74 (1989; Zbl 0717.34004)] where the following result was proved. Assume that a system of ordinary differential equations has a solution which is a compound asymptotic. Then there exists a smooth solution with the same compound asymptotic. Thus, the goal \({\mathbf G}_ 4\) from the item 4.8 of the paper under review is partially achieved in the case of differentiable functions in one real variable.

For the entire collection see [Zbl 0780.00040].

One can divide this manuscript into two parts. The first part consists of three lectures where basic notations, definitions and results from previous author’s works are described [see the author, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques, Hermann, Paris. (1992; Zbl 1241.34003)] and [Compensation of small denominators and ramified linearization of local objects’, Astérisque V. 222, 135–199 (1994; Zbl 0810.58036)]. In particular the notions of resurgent functions, alien calculus, medianization, compensation and seriation, moulds and comoulds, arborification are discussed. Then the three main facts concerning local objects, in other words, germs of vector fields or diffeomorphisms of \(\mathbb{C}^ m\) are recalled. Thus, it is well known that local objects are formally linearized in the non-resonance case. Moreover if such an object isn’t quasiresonance then it can be analytically linearized. In the quasiresonance case the convergence of the formal linearizators may be restorted by means of the arborification. Any nondegenerate local object can be linearized by means of a seriable- ramified change of coordinates, that is, by substitutions \(y_ i = x_ i(1 + \varphi_ i (x))\) which are formal series in variables \(x_ i, \log x_ i\). Above all it is also supposed that such a series must be summable in a unique way, by means of suitable Borel-Laplace procedure, to a sum that is defined and analytic in a ramified slowly spiraling neighborhood of the origin in \(\mathbb{C}_ \bullet^ m\) where \(\mathbb{C}_ \bullet\) is the Riemann surface of the logarithm.

The second part also consists of three lectures. A construction of the fields of transseries and analysable germs is described in detail. Transseries are series of transmonomials, that is, elements of the scale of infinity in the sense of Bourbaki. The basis of the scale contains \(z^ \sigma\), \(\sigma \in \mathbb{R}\), and it is closed under the log- and exp-operations. The algebra of transseries \(\mathbb{R}[[[z]]]\) may be characterized as being the smallest algebra that i) contains \(z\) and \(\mathbb{R}\); ii) is stable under the precomposition by exp-operation and the postcomposition by log-operation; iii) contains \(\mathbb{R} [[\varepsilon_ 1, \dots, \varepsilon_ r]]\) as soon as it contains transseries \(\varepsilon_ 1, \dots, \varepsilon_ r\) (for \(\varepsilon_ i (\infty) = 0\), formally). The next challenge would be to prove that formal solutions of any analytic dynamical system have canonical analysable expansions. A ramified analytic germ having asymptotic transseries is called the analysable function. The procedure of summation of transseries into an analysable germ is known as the accelero-synthesis, and reverse procedure is known as the decelero-analysis. This procedure consists in iterative executing of the following five steps: 1) changing of independent variables to sufficiently low ones; 2) carrying out the formal Borel transform; 3) changing of the Borel variable to the corresponding fast variables with the aid of an integral transformation; 4) performing the Laplace transformation; 5) reverting to the original variable. The summation is a constructive correspondence which is compatible with the canonical embedding. A proposed simple and conceptual proof of non-accumulation of limit cycles of real-analytic, first-order differential equations is based on the last properties. The return map turns out to be analysable and hence it is either identical or monotonic as well as its transseries.

It should be remarked that a similar theory of transseries called compound asymptotics has been developed in an earlier work [A. N. Kuznetsov, Funct. Anal. Appl. 23, No. 4, 308–314 (1989); translation from Funkts. Anal. Prilozh. 23, No. 4, 63-74 (1989; Zbl 0717.34004)] where the following result was proved. Assume that a system of ordinary differential equations has a solution which is a compound asymptotic. Then there exists a smooth solution with the same compound asymptotic. Thus, the goal \({\mathbf G}_ 4\) from the item 4.8 of the paper under review is partially achieved in the case of differentiable functions in one real variable.

For the entire collection see [Zbl 0780.00040].

Reviewer: A.G.Aleksandrov (Moskva)

### MSC:

32S65 | Singularities of holomorphic vector fields and foliations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34C99 | Qualitative theory for ordinary differential equations |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

44A10 | Laplace transform |