Lyapunov graphs, Poincaré-Hopf and Morse inequalities

Lyapunov graphs carry dynamical information of gradient-like fows as well as topological information of its phase space which is taken to be a closed orientable n-manifold.In this article we will show that an abstract Lyapunov graph L(h0,..., hn,k) in dimension n greater than two, with cycle number k, satisfies the Poincaré-Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number g 1³ k. We also show a continuation theorem for abstract Lyapunov graphs with the presence of cycles. Finally, a family of Lyapunov graphs L(h0,..., hn,k) with fixed pre-assigned data (h0,..., hn,k) is associated with the Morse polytope, Pk(h0,..., hn), determined by the Morse inequalities for the given data.