Un indice qui affine l’indice de Poincaré-Lefschetz pour les homeomorphismes de surfaces

Abstract

Nous étudions la dynamique des homéomorphismes de surfaces autour des points fixes isolés dont l’indice de Poincaré-Lefschetz est différent de $1$. Nous définissons un invariant de conjugaison, mot cyclique sur l’alphabet $\{\uparrow, \rightarrow , \downarrow , \leftarrow\}$, qui affine l’indice de Poincaré-Lefschetz. Grossièrement, il s’agit de décomposer canoniquement la dynamique en un nombre fini de secteurs hyperboliques, elliptiques ou indifférents, chaque type de secteur contribuant à l’indice de Poincaré-Lefschetz par un terme valant respectivement $-1/2$, $+1/2$ ou $0$. Le mot cyclique permet de lire l’existence de structures dynamiques canoniques.

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Authors

Frédéric Le Roux

Laboratoire de Mathématiques
Université Paris Sud
Bâtiment 425/430
91405 Orsay Cedex
France