Lusternik – Schnirelman Theory and Dynamics

In this paper we study a new topological invariant Cat(X, ξ), where X is a finite polyhedron and ξ ∈ H(X;R) is a real cohomology class. Cat(X, ξ) is defined using open covers of X with certain geometric properties; it is a generalization of the classical Lusternik – Schnirelman category. We show that Cat(X, ξ) depends only on the homotopy type of (X, ξ). We prove that Cat(X, ξ) allows to establish a relation between the number of equilibrium states of dynamical systems and their global dynamical properties (such as existence of homoclinic cycles and the structure of the set of chain recurrent points). In the paper we give a cohomological lower bound for Cat(X, ξ), which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.