Un indice qui affine l'indice de Poincaré-Lefschetz pour les homéomorphismes de surfaces

Frédéric Le Roux (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We study the dynamics of surface homeomorphisms around isolated fixed points whose Poincaré-Lefschetz index is not equal to $1$. We construct a new conjugacy invariant, which is a cyclic word on the alphabet $\{\ua, \ra , \da , \la\}$. This invariant is a refinement of the P.-L. index. It can be seen as a canonical decomposition of the dynamics into a finite number of sectors of hyperbolic, elliptic or indifferent type. The contribution of each type of sector to the P.-L. index is respectively $-1/2$, $+1/2$ and $0$. The construction of the invariant implies the existence of some canonical dynamical structures.

Mots clés: Homeomorphism; surface; fixed point; Brouwer; index; Reeb components

Domaines: Systèmes dynamiques [math.DS]

Frédéric Le Roux. Un indice qui affine l'indice de Poincaré-Lefschetz pour les homéomorphismes de surfaces. 2005. ⟨hal-00005108⟩