  We prove a non-mixing property of the flow of the 3D Euler equation which has
a local nature: in any neighbourhood of a "typical" steady solution there is a
generic set of initial conditions, such that the corresponding Euler flows will
never enter a vicinity of the original steady one. More precisely, we establish
that there exist stationary solutions $u_0$ of the Euler equation on $\mathbb
S^3$ and divergence-free vector fields $v_0$ arbitrarily close to $u_0$, whose
(non-steady) evolution by the Euler flow cannot converge in the $C^k$ H\"older
norm ($k>10$ non-integer) to any stationary state in a small (but fixed a
priori) $C^k$-neighbourhood of $u_0$. The set of such initial conditions $v_0$
is open and dense in the vicinity of $u_0$. A similar (but weaker) statement
also holds for the Euler flow on $\mathbb T^3$. Two essential ingredients in
the proof of this result are a geometric description of all steady states near
certain nondegenerate stationary solutions, and a KAM-type argument to generate
knotted invariant tori from elliptic orbits.
