Compactly Generated Shape Index for Infinite-dimensional Local Dynamical Systems on Complete Metric Spaces

In this paper, we establish a theory of compactly generated shape index for local semiflows on complete metric spaces via very ordinary shape index pairs. The main advantage is that the quotient space $N/E$ is not necessary to be metrizable for the shape index pair $(N,E)$. In the new shape theory, we can calculate the shape index of a compact isolated invariant set $K$ in any closed subset that contains a local unstable manifold of $K$, and define the shape cohomology index of $K$ to develop the Morse equations. This provides more effective ways to calculate shape indices and Morse equations theoretically and specifically for infinite dimensional systems, without particular requirements on the index pairs or the unstable manifolds.