Lusternik–Schnirelman Theory and Dynamics. II

We show how the methods of homotopy theory can be used in dynamics to study the topology of a chain recurrent set. More specifically, we introduce new homotopy invariants $\mathrm {cat}^1(X,\xi)$ and $\mathrm {cat}^1_{\mathrm s}(X,\xi)$ that depend on a finite polyhedron $X$ and a real cohomology class $\xi \in H^1(X;\mathbb R)$ and are modifications of the invariants introduced earlier by the first author. We prove that, under certain conditions, $\mathrm {cat}_{\mathrm s}^1(X,\xi)$ provides a lower bound for the Lusternik–Schnirelman category of the chain recurrent set $R_\xi$ of a given flow. The approach of the present paper applies to a wider class of flows compared with the earlier approach; in particular, it allows one to avoid certain difficulties when checking assumptions.