Lusternik - Schnirelman theory and dynamics

Details Lusternik - Schnirelman theory and dynamics Lusternik - Schnirelman theory and dynamics

2002-04-11

arxiv.org/abs/math/0204149v1

Mathematics

Algebraic Topology

20 pages

Scientific paper

In this paper we study a new topological invariant $\Cat(X,\xi)$, where $X$ is a finite polyhedron and $\xi\in H^1(X;\R)$ is a real cohomology class. $\Cat(X,\xi)$ is defined using open covers of $X$ with certain geometric properties; it is a generalization of the classical Lusternik -- Schnirelman category. We show that $\Cat(X,\xi)$ depends only on the homotopy type of $(X,\xi)$. We prove that $\Cat(X,\xi)$ 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 $\Cat(X,\xi)$, which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.

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