Well-behaved convolution averages and the non-accumulation theorem for limit-cycles. (English) Zbl 0857.34009

Ecalle, Jean; Menous, Frédéric

Braaksma, B. L. J. (ed.) et al., The Stokes phenomenon and Hilbert’s 16th problem. Proceedings of the workshop, Groningen, The Netherlands, May 31-June 3, 1995. Singapore: World Scientific. 71-101 (1996).

Summary: This expository paper introduces several uniformizing averages, which are serviceable in resummation theory because they manage to reconcile three essential, but at first sight quite conflicting demands: respecting convolution; preserving realness; reproducing latteral growth. Their potential range of application covers most situations characterized by a combination of (1) nonlinearity; (2) divergence, (3) realness. We sketch three typical applications, the last of which leads to a marginal, yet significant simplification of the constructive (i.e. resummation-theoretical) proof of the non-accumulation theorem for the limit-cycles of a real-analytic vector field on \(\mathbb{R}^2\).
For the entire collection see [Zbl 0846.00026].

MSC:

34M99	Ordinary differential equations in the complex domain
32S05	Local complex singularities
32S65	Singularities of holomorphic vector fields and foliations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
40A99	Convergence and divergence of infinite limiting processes
40H05	Functional analytic methods in summability
53B99	Local differential geometry
37-XX	Dynamical systems and ergodic theory
60G15	Gaussian processes
60J60	Diffusion processes
60J65	Brownian motion

Keywords:

expository paper; uniformizing averages; convolution; realness; latteral growth; non-accumulation; limit-cycles