The geometry of cross sections to flows.

*(English)*Zbl 0594.58041A cross section K to a flow \(\phi\) on M is a closed codimension 1 submanifold which intersects transversely all flow lines. A more intrinsic point of view is to consider an appropriate \({\mathbb{Z}}\)-cover \(\tilde M\) such that \(\phi\) lifts to \({\tilde \phi}\) and the orbit space \(\tilde M/{\mathbb{R}}\) is an external cross section of \(\phi\). Simple necessary and sufficient conditions are given under which an external cross section can be imbedded in M as a cross section.

Consider \(D_ M=H_ 1(M;{\mathbb{R}})/{\mathbb{R}}^+\); this gives a sphere of all nonzero directions in \(H_ 1(M;{\mathbb{R}})\) and a point. The set \(D_{\phi}\) of homology directions of \(\phi\) is defined. A homology direction roughly corresponds to the normalized homology class of a long almost closed orbit. Every cross section K determines a class \(u_ K\in H^ 1(M;{\mathbb{Z}})\) \((u_ K(\ell)\), where \(\ell\) is an oriented loop, is the intersection number of K and \(\ell).\)

Two of the author’s main results are the following. Theorem C: Let K and L be cross sections and \(u_ K=u_ L\); then there is a smooth family of cross sections \(K_ t\) with \(K_ 0=K\) and \(K_{\ell}=L\). Theorem D: Let \(u\in H^ 1(M;{\mathbb{Z}})\); then \(\phi\) has a cross section in class u if and only if u is positive on \(D_{\phi}\). Homology directions are computed for Axiom A flows, which gives a finite criterion for the existence of cross sections. The behavior of homology directions under perturbation turns out to be rather hectic (at least if there are no cross sections). Homology directions for a suspension flow are computed and the relations between minimal sets, the Birkhoff center and the existence of cross sections are studied in the paper.

Consider \(D_ M=H_ 1(M;{\mathbb{R}})/{\mathbb{R}}^+\); this gives a sphere of all nonzero directions in \(H_ 1(M;{\mathbb{R}})\) and a point. The set \(D_{\phi}\) of homology directions of \(\phi\) is defined. A homology direction roughly corresponds to the normalized homology class of a long almost closed orbit. Every cross section K determines a class \(u_ K\in H^ 1(M;{\mathbb{Z}})\) \((u_ K(\ell)\), where \(\ell\) is an oriented loop, is the intersection number of K and \(\ell).\)

Two of the author’s main results are the following. Theorem C: Let K and L be cross sections and \(u_ K=u_ L\); then there is a smooth family of cross sections \(K_ t\) with \(K_ 0=K\) and \(K_{\ell}=L\). Theorem D: Let \(u\in H^ 1(M;{\mathbb{Z}})\); then \(\phi\) has a cross section in class u if and only if u is positive on \(D_{\phi}\). Homology directions are computed for Axiom A flows, which gives a finite criterion for the existence of cross sections. The behavior of homology directions under perturbation turns out to be rather hectic (at least if there are no cross sections). Homology directions for a suspension flow are computed and the relations between minimal sets, the Birkhoff center and the existence of cross sections are studied in the paper.