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 Hhyp(2g − 2) and Hhyp(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 Hhyp(2g−2) or Hhyp(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.