Counting invariant components of hyperelliptic translation surfaces

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 H (hyp) (2g - 2) and 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 H (hyp) (2g-2) or H (hyp) (g-1, g-1) with precisely p periodic components and m minimal components. This extends results by Naveh ([Nav08]), who obtained tight upper bounds on numbers of invariant components for each stratum.