Groups of basic automorphisms of chaotic Cartan foliations with Eresmann connection

N. I. Zhukova ; K. I. Sheina

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 6, pp. 897-907

Résumé

The purpose of the work is to study the groups of basic automorphisms of chaotic Cartan foliations with Ehresmann connection. Cartan foliations form a category where automorphisms preserve not only the foliation, but also its transverse Cartan geometry. The group of basic automorphisms of a foliation is the quotient group of the group of all automorphisms of this foliation by the normal subgroup of leaf automorphisms with respect to which each leaf is invariant. Cartan foliations include such wide classes of foliations as pseudo-Riemannian, Lorentzian, and foliations with transversal affine connection. No restrictions are imposed on the dimension of either the foliation or the foliated manifold. Compactness of the foliated manifold is not assumed. Methods. The proof of the structure theorem for chaotic Cartan foliations is based on the application of the foliated bundle construction, commonly used in the theory of foliations with transverse geometries. Results. The main result of this paper is the theorem stating that the group of basic automorphisms of any chaotic Cartan foliation with Ehresmann connection admits the structure of a Lie group and finding estimates for the dimension of this group. In particular, it is proved that if the set of closed leaves is countable, then the group of basic automorphisms of such a foliation is countable. Conclusion. In this paper, we prove a criterion according to which the chaoticity of a Cartan foliation of type $(G, H)$ is equivalent to the chaoticity of a locally free action of the group $H$ on the associated parallelizable manifold. Thus, the problem of the existence of chaos in Cartan foliations with Ehresmann connection reduces to the same problem for locally free actions of a Lie group on parallelizable manifolds.  

Détail

Keywords: foliation, Ehresmann connection for foliation, chaotic foliation, basic automorphism of a foliation