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This paper is concerned with the Morse theory of attractors for semiflows oncomplete metric spaces. First, we construct global Morse-Lyapunov functions forMorse decompositions of attractors. Then we extend some well known deformationresults in the critical-point theory to Morse-Lyapunov functions which are onlycontinuous. Based on these works, we finally introduce the concept of criticalgroups for Morse sets and establish Morse inequalities and Morse equations forattractors.

Morse Theory of Attractors via Lyapunov Functions