数学科学学院学术报告
——几何测度论与相关领域讨论班
Formal normalization and formal invariant foliation for an elliptic fixed point in the plane
(joint work with Alain Chenciner (Paris), Shanzhong Sun and Qiaoling Wei (CNU))
David Sauzin 研究员
（法国国家科研中心-IMCCE 巴黎天文台）
报告时间：2024年11月26日 (星期二) 下午14:00
报告地点: 学院路校区新主楼F122
报告摘要：Classically, for a local analytic diffeomorphism $F$ of $(R^2,0)$ with a non-resonant elliptic fixed point (eigenvalues $\exp(\pm 2\pi i\omega)$ with $\omega$ real irrational), one can find formal normalizations, i.e. formal conjugacies to a formal diffeomorphism invariant under the group of rotations. Less demanding is the notion of a "geometric normalization" that we introduce: this is a formal conjugacy to a formal diffeomorphism which maps any circle centered at $0$ to a circle centered at $0$. Geometric normalizations are not unique, but they correspond in a natural way to a unique formal invariant foliation (any leaf is mapped to a leaf by $F$). Suppose that $\omega$ is super-Liouville. We then show that, generically, all geometric normalizations are divergent, so there is no analytic invariant foliation. This is a sequel—or rather a prequel—to [A. Chenciner, D. Sauzin, S. Sun & Q. Wei: Elliptic fixed points with an invariant foliation: Some facts and more questions, RCD 2022, Vol. 27], which will be reviewed too: in the exceptional situation where $F$ leaves invariant an analytic foliation, formal normalizations are still generically divergent.
报告人简介：Dr. Sauzin is a senior researcher in mathematics at CNRS in France. He has worked intensively in the areas of nonlinear dynamical systems, summability theory, Ecalle's resurgence theory (which deals primarily with the behavior of asymptotic series or transseries), and mould calculus (a powerful combinatorial technique). He contributed to the theory of Hamiltonian perturbations (exponentially small separatrix splitting, Nekhoroshev's exponential stability theorem and examples of Arnold diffusion and wandering domains in Gevrey near-integrable systems), averaging theory in Gevrey classes, and holomorphic dynamics in one or two dimensions, and more recently applications of Resurgence to mathematical physics (Deformation quantization, quantum modularity and TQFT). He has made very fundamental contributions to Summability theory and Resurgence theory and is member of M. Kontsevich's team within the ERC Synergy Grant "Recursive and Exact New Quantum Theory" project.
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