On the symplectic phase space of KdV

We prove that the Birkhoff map \Om for KdV constructed on H^{-1}_0(\T) can be interpolated between H^{-1}_0(\T) and L^2_0(\T). In particular, the symplectic phase space H^{1/2}_0(\T) can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential q\in H^{-1}(\T) in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$

Lyapunov 1-forms for flows

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.Comment: 27 pages, 3 figures. This revised version incorporates a few minor improvement