Formal normalization and formal invariant foliation for an elliptic fixed point in the plane
数学专题报告
报告题目(Title)：Formal normalization and formal invariant foliation for an elliptic fixed point in the plane (系列报告2)
报告人(Speaker)：Prof. David Sauzin（巴黎天文台）
地点(Place)：后主楼 1220
时间(Time)：2024年11月13日 19:00-20:00
邀请人(Inviter)：苏喜锋
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 to: in the exceptional situation where $F$ leaves invariant an analytic foliation, formal normalizations are still generically divergent. This is a joint work with Alain Chenciner (Paris), Shanzhong Sun and Qiaoling Wei (CNU).
