  We study the horocycle flow on the stratum of translation surfaces
$\mathcal{H}(2)$, showing that there is a sequence of horocycle ergodic
measures, supported on a sequence of periodic horocycle orbits, which weak-$*$
converges to an $\mathrm{SL}_2(\mathbb{R})$-invariant but not ergodic measure.
As a consequence, we show that there are points in $\mathcal{H}(2)$ that do not
equidistribute under the horocycle flow for any measure.
