  We construct finite dimensional families of non-steady solutions to the Euler
equations, existing for all time, and exhibiting all kinds of qualitative
dynamics in the phase space, for example: strange attractors and chaos,
invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of
any dimension.
  The main theorem of the paper, from which these families of solutions are
obtained, states that for any given vector field $X$ on a closed manifold $N$,
there is a Riemannian manifold $M$ on which the following holds: $N$ is
diffeomorphic to a finite dimensional manifold in the phase space of fluid
velocities (the space of divergence-free vector fields on $M$) that is
invariant under the Euler evolution, and on which the Euler equation reduces to
a finite dimensional ODE that is given by an arbitrarily small perturbation of
the vector field $X$ on $N$.
