Lyapunov graphs and inequalities of Poincaré-Hopf and Morse

Author(s)

Maria Alice Bertolim

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2002-11-29

Examining board members

Ketty Abaroa de Rezende; Maria do Carmo Carbinatto; Maria Herminia de Paula Leite Mello; Caio Jose Colletti Negreiros; Marco Antonio Teixeira

• Advisor: Ketty Abaroa de Rezende; Oziride Manzoli Neto

Abstract

Lyapunov graphs carry dynamical information of gradient-like flows as well as topological information of its phase space which is taken to be a closed orientable n-manifold. In this thesis the Lyapunov graphs L(h0,..., hn, K) considered may represent smooth flows on closed orientable n-manifolds, n ≥ 2, with cycle number K. We will show that the Poincaré-Hopf inequalities are necessary and sufficient conditions for an abstract Lyapunov graph L(h0,..., hn, K) to be continued to an abstract Lyapunov graph of Morse type with cycle rank greater or equal to K. The continuation which is presented by means of a constructive algorithm, is shown to be unique in dimensions two and three. In ali other dimensions, the exact number of possible continuations of L are presented. We show that an abstract Lyapunov graph L(h0,..., hn, K) in dimension n greater than or equal to two, with cycle number K, satisfies the Poincaré-Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number γ1 is at least equal to K. The convex hull of the collection of ali Betti number vectors which satisfy the Morse inequalities and the inequality γ 1 ≥ K for a pre-assigned data determines a Morse polytope PK (h0....,hn). Finally, we associate a Morse polytope, PK..., hn) to a family of Lyapunov graphs L(h0 ... hh, K) and determine geometrical properties of this polytope. (AU)