Time: Oct 17, 24, 31, Nov 7, 14, 21 13:30-16:15
Place: 教二112
Speaker: D. Sauzin (CNRS, LTE-Paris Observatory)
Title: Tree expansion methods for dynamical systems
Abstract: "Armould Calculus" is a combinatorial apparatus developed by French mathematician Jean Ecalle with a view to nonlinear dynamics problems. Although not yet very well-known in the mathematical community, it has potential for a number of applications. This minicourse aims at making these tools understandable to the students, with a view to getting them acquainted with Dynamical Systems topics that are classically deemed "hard analysis".
The application we will focus on is the linearization problem for local analytic nonresonant dynamical systems in any number of dimensions, with continuous or discrete time. The nonresonance condition makes it possible to find a formal solution, but unavoidable near-resonances make its convergence difficult to tackle. Ecalle's Armould Calculus yields an efficient non-inductive representation of the formal solution, which allows one to get a quantitative version of the classical Bruno-Russmann theorems: convergence can be established under the Bruno condition (this arithmetical condition is known to be the optimal one in dimension one, thanks to J.-C. Yoccoz's celebrated work on the normalization of quadratic polynomials), with small denominators controlled well enough to to get the best known estimates for the domain of convergence of the solution.
We will introduce the armould formalism by providing all the necessary definitions in terms of decorated trees. It is related to the Connes-Kreimer Hopf algebra as well as Grossman-Larson and shuffle/quasishuffle Hopf algebras. In the context of the linearization problem for multi-dimensional germs and vector fields, this combinatorial framework leads to beautiful algebraic cancellations, which reduce the analytical work to little and keep everything conceptual.
References:
J. Ecalle, "Singularities that are inaccessible by geometry," Ann. Inst. Fourier (Grenoble) 42, 1–2 (1992), 73–164.
J. Ecalle, B. Vallet, "The arborification–coarborification transform: analytic, combinatorial, and algebraic aspects", Ann. Fac. Sci. Toulouse Math. (6) 13, 4 (2004), 575–657.
F. Fauvet, F. Menous, D. Sauzin, "Explicit linearization of one-dimensional germs through tree-expansions"
Bull. soc. math. France, 146, 2 (2018), pp.241-285. https://hal.science/hal-01053805v2
F. Fauvet, F. Menous, D. Sauzin, "Explicit linearization of multi-dimensional germs and vector fields through Ecalle’s tree expansions", Preprint, 2025. https://arxiv.org/abs/2507.13216
