On the Morse theory of attractors: A functional approach. (November 2021) - Li, Desheng; Jia, Mo
Abstract: This paper aims at developing a functional approach to investigate the Morse structures of attractors for infinite-dimensional dynamical systems generated by PDEs. Let S ( t ) be a semiflow on a complete metric space X, and A an attractor of S ( t ) with a Morse decomposition M . We first construct some good Morse–Lyapunov functions for M which seem to be of independent interest in their own right. Then we establish some fundamental deformation lemmas. Based on these results, we finally introduce the concept of critical groups for Morse sets and prove Morse inequalities and equations. The problem of how to calculate Morse equations of attractors by using a natural Morse–Lyapunov function which may be defined only on a dense subspace of X is also addressed, and an illustrating example of parabolic equation is presented.