Counting invariant components of hyperelliptic translation surfaces

Abstract: 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.