Study of KdV hierarchy from ergode theory

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Segal- Wilson’s interpretation of Sato’s grassmann manifold method was applied to the closure of the space of all reflectionless potentials, and a KdV flow on it was constructed. In the construction a notion of positivity of subspaces was introduced and used to show the non-degeneracy of the τ-function. Remling’s theorem on shift flow was extended to the KdV flow by introducing generating functions and monodromy matrices describing the group action on the generating functions, which made it possible to characterize invariant subsets of the KdV flow.