Chaos in topological foliations

N. I. Zhukova ; G. S. Levin ; N. S. Tonysheva

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 68 (2022) no. 3, pp. 424-450

Résumé

We call a foliation $(M, F)$ on a manifold $M$ chaotic if it is topologically transitive and the union of closed leaves is dense in $M.$ A foliated manifold $M$ is not assumed to be compact. The chaotic foliations can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. We introduce the concept of the integrable Ehresmann connection for a foliation as a natural generalization of the integrable Ehresmann connection for smooth foliations. A description of the global structure of foliations with integrable Ehresmann connection and a criterion for the chaotic behavior of such foliations are obtained. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic foliations of codimension two on $3$-dimensional closed and nonclosed manifolds is constructed.