  Understanding complexity in fluid mechanics is a major problem that has
attracted the attention of physicists and mathematicians during the last
decades. Using the concept of renormalization in dynamics, we show the
existence of a locally dense set $\mathscr G$ of stationary solutions to the
Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$
is universal in the sense that any area preserving diffeomorphism of the disk
can be approximated (with arbitrary precision) by the Poincar\'e map of $X$ at
some transverse section. We remark that this universality is approximate but
occurs at all scales. In particular, our results establish that a steady Euler
flow may exhibit any conservative finite codimensional dynamical phenomenon;
this includes the existence of horseshoes accumulated by elliptic islands,
increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic
tangencies of arbitrarily high multiplicity. The steady solutions we construct
are Beltrami fields with sharp decay at infinity. To prove these results we
introduce new perturbation methods in the context of Beltrami fields that allow
us to import deep techniques from bifurcation theory: the
Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte
theorems on the geometry of wild hyperbolic sets. These perturbation methods
rely on two tools from linear PDEs: global approximation and
Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I.
Arnold's vision on the complexity of Beltrami fields in Euclidean space.
