Tangent to the Identity Germs and Affine Surfaces

Parabolic manifolds of analytic diffeomorphisms along an invariant formal curve

13 févr. 2025, 14:00

50m

Amphithéatre Schwartz (Institut de Mathématiques de Toulouse)

Orateur

Fernando Sanz

Description

Let $F:({\mathbb{C}}^n,0)\to ({\mathbb{C}}^n,0)$ be a germ of a holomorphic diffeomorphism and let $\mathrm{\Gamma }$ be a formal curve at $0$, invariant for $F$. Under certain sharp conditions on the restricted diffeomorphism $F{|}_{\mathrm{\Gamma }}$, we show that there exists a finite non-empty family of complex submanifolds of ${\mathbb{C}}^n\setminus \{0\}$, invariant for $F$ and entirely composed of orbits which converge to the origin and have flat contact with $\mathrm{\Gamma }$ (parabolic manifolds). In a second part of the talk, we adapt this result for the case of a germ of a real analytic diffeomorphism $F:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$, where we can show, moreover, that each parabolic manifold of the family is foliated by real parabolic curves of $F$.

These results are obtained in collaboration with L. López-Hernánz, J. Ribón, J. Raissy and L. Vivas.