Chaos in topological foliations (Q6562949)

The purpose of the paper under review is the study of the chaotic behavior of topological foliations in manifolds of any dimension and not necessarily compact, following the parallelism with what other authors have been developing in the case of smooth foliations. Consider an \(n\)-dimensional topological manifold \(M\) and a topological foliation \(F\) of codimension \(q\) (\(1\leq q \leq n\)) on \(M\). In the first part of the paper (sections \(\S1\) to \(\S5\)) the authors study the chaotic behaviour of \(F\) produced if there is a dense leaf of \(F\) and the union of the closed leaves of \(F\) is dense in \(M\). The main result is a characterization of such a chaotic foliation \(F\) when the universal covering map \(k:\widetilde{M}\to M\) induces a foliation \(\widetilde{F}\) on \(\widetilde{M}\) whose \((n-q)\)-dimensional leaves are locally given by the fibres of some locally trivial bundle \(r:\widetilde{M}\to \widetilde{T}\) over a \(q\)-dimensional topological manifold \(\widetilde{T}\). They prove that \(F\) is chaotic (in the above sense) if and only if the (global) holonomy group of \(F\) (as a subgroup of Homeo\((\widetilde{T})\)) has an element whose orbit is dense and the union of all closed orbits is dense (Theoren 5.2).\N\NIn the second part of the paper (sections \(\S6\) to \(\S8\)), the authors study a topological foliation \(F\) with a \(q\)-dimensional integrable distribution \(\mathfrak{M}\) transversal to \(F\). They introduce the notion of an integrable Ehresmann connection for \(F\) as the topological foliation \(F_{t}\) on \(M\) of dimension \(q\) associated with \(\mathfrak{M}\) in case that \(\widetilde{M}\) is a product \(\widetilde{M}_{1}\times \widetilde{M}_{2} \) and the foliations \(\widetilde{F}\) and \(\widetilde{F}_{t}\) (induced by the universal covering map \(k\)) on \(\widetilde{M}\) are the corresponding trivial foliations of the product \(\widetilde{M}_{1}\times \widetilde{M}_{2} \). With this definition they obtain a representation of a foliation \(F\) with an integrable Ehresmann connection \(F_{t}\) as a canonical foliation defined on a product of two manifolds \(X\times T\) modulo the action of a group of homeomorphisms \(\Psi\) on it (Theorem 6.2), and thus conclude that such a foliation \(F\) is chaotic when the group \(\Psi\) on \(T\) has an element whose orbit is dense and the union of all closed orbits is dense (Theorem 6.3). Finally, the authors also apply these results to study topological chaotic suspended foliations on \(3\)-manifolds.\N\NThis paper contains interesting results about topological foliations covered by fibrations.