Chaos in Topological Foliations

Journal of Mathematical Sciences. 2024. Vol. 282. No. 3. P. 337–361.

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

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. The 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 the 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.

Keywords: fibration, holonomy group, chaotic foliation, foliated manifold, integrable Ehresmann connection