  In this paper we address the Cauchy problem for the incompressible Euler
equations in the periodic setting. Based on estimates developed in
[Buckmaster-De Lellis-Isett-Sz\'ekelyhidi], we prove that the set of H\"older
$1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial
datum wild if it admits infinitely many admissible H\"older $1\slash 5-\eps$
weak solutions. We also introduce a new set of stationary flows which we use as
a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds
stresses.
