Poincaré–Hopf and Morse inequalities for Lyapunov graphs

Abstract

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 $L(h_0, \dotsc,h_n,\kappa)$ in dimension n greater than 2, with cycle number $\kappa$, satisfies the Poincaré–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 $\mathcal{L}(h_0, \dotsc, h_n,\kappa)$ with fixed pre-assigned data $(h_0, \dotsc, h_n,\kappa)$ is associated with the Morse polytope, $\mathcal{P}_{\kappa}(h_0,\dotsc, h_n)$, determined by the Morse inequalities for the given data.