Abstract

We show how the methods of homotopy theory can be used in dynamics to study the topology of chain recurrent sets. More precisely, we introduce the new homotopy invariants cat1(X,ξ) and cat1s(X,ξ) for a given finite polyhedron X and a cohomology class ξ∈H1(X;R), which are modifications of the invariants introduced earlier by the first author. We prove that, under certain conditions, cat1s(X,ξ) provides a lower bound for the Lyusternik-Shnirelʹman category of the chain recurrent set Rε of a given flow. The approach of the present paper applies to a wider class of flows than the earlier approach; in particular, it avoids certain difficulties involved in checking the assumptions.