On The Theory Of Conley Index And Shape Conley Index In General Metric Spaces

It is mainly studied in this paper the Conley index and shape index of compact isolated invariant sets of local dynamical systems on non-compact metric spaces.First, we establish the local dynamical systems on general topological spaces and its attractor theory, and give the existence of the Lyapunov functions of at-tractors and the Morse decomposition of sequential compact invariant sets. Then,from the view of attractor theory, we provide a new frame on the Conley index theory of infinite dimensional dynamical systems, by which this theory is greatly simplified. As to the shape index, we introduce a new shape Conley index pair,and based on the attractor theory of topological dynamical systems we construct the shape Conley index theory and Morse theory. This work generalizes the con-sequences on locally compact spaces to the case of non-locally compact spaces,and it makes the calculation and theoretical analysis of the shape index much more convenient and flexible.