Cocycles and Ma~{n}e sequences with an application to ideal fluids

Exponential dichotomy of a strongly continuous cocycle $bFi$ is proved to be equivalent to existence of a Ma~{n}e sequence either for $bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.