Morse–Smale theory for flows on covering spaces of compact manifolds

Abstract

For a dynamical system X on a compact differentiable manifold M and for the dynamical system X(ρ) induced from X by a covering map \({\rho \, : \, \widetilde{M}\, \rightarrow \, M}\), we develop algebraic topology methods for estimating the lower bounds on the number of codimension-1 surfaces (i.e., on the number of index-1 equilibria of flows and their stable manifolds) on the boundary of regions of stability on \({\widetilde{M}}\). We also develop methods for estimating the number of equilibria on the boundaries of stability regions of noncompact manifolds with very general assumptions. Our methods allow us to obtain results for noncompact manifolds in cases when Morse–Smale approach does not work.