Global attractors of complete conformal foliations

N. I. Zhukova

Sbornik. Mathematics, Tome 203 (2012) no. 3, pp. 380-405

Résumé

We prove that every complete conformal foliation $(M,\mathscr F)$ of codimension $q\geqslant 3$ is either Riemannian or a $(\operatorname{Conf}(S^q), S^q)$-foliation. We further prove that if $(M,\mathscr F)$ is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In this theorem we do not assume that the manifold $M$ is compact. In particular, every proper conformal non-Riemannian foliation $(M,\mathscr F)$ has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all nonclosed leaves is a connected $q$-dimensional orbifold. We show that every countable group of conformal transformations of the sphere $S^q$ can be realized as the global holonomy group of a complete conformal foliation. Examples of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed. Bibliography: 20 titles.

Détail

Keywords: global holonomy group, minimal set, global attractor.
Mots-clés : conformal foliation