Poincare-Hopf and Morse inequalities for Lyapunov graphs

Lyapunov graphs carry dynamical information of gradient-like flows as well as topological information of their phase space which is taken to be a closed orientable n-manifold. In this paper we will show that an abstract Lyapunov graph L(h(0), . . . , h(n), kappa) in dimension n greater than 2, with cycle number kappa, satisfies the Poincare-Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number, gamma(1), is greater than or equal to kappa. We also show a continuation theorem for abstract Lyapunov graphs with the presence of cycles. Finally, a family of Lyapunov graphs L(h(0), . . . , h(n), kappa) with fixed pre-assigned data (h(0), . . . , h(n), kappa) is associated with the Morse polytope, P-kappa(h(0), . . . , h(n)), determined by the Morse inequalities for the given data.