Chaos in the incompressible Euler equation on manifolds of high dimension

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