  The flow in a fixed direction on a translation surface S determines a
decomposition of S into closed invariant sets, each of which is either periodic
or minimal. We study this decomposition for translation surfaces in the
hyperelliptic connected components $\mathcal{H}^{hyp}(2g-2)$ and
$\mathcal{H}^{hyp}(g-1,g-1)$ of the corresponding strata of the moduli space of
translation surfaces. Specifically, we characterize the pairs of nonnegative
integers (p,m) for which there exists a translation surface in
$\mathcal{H}^{hyp}(2g-2)$ or $\mathcal{H}^{hyp}(g-1,g-1)$ with precisely p
periodic components and m minimal components. This extends results by Naveh
([Naveh08]), who obtained tight upper bounds on the numbers of minimal
components and invariant components a translation surface in any given stratum
may have. Analogous results for the other connected components of moduli space
are forthcoming.

Dynamical Systems