Abstract
In this paper we study a new topological invariant Cat(X,�), where X is a finite polyhedron and � 2 H 1 (X; R) is a real cohomology class. Cat(X,�) 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,�) depends only on the homotopy type of (X,�). We prove that Cat(X,�) 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,�), which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
