Lengths spectrum of hyperelliptic components

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmuller geodesic flow restricted to the hyperelliptic connected components of the strata of Abelian differentials, settling a question of Farb [Problems on Mapping Class Groups and Related Topics, 11-55 (2006)]. These are the first results on the description of the systole of moduli spaces, outside some cases in low genera. We emphasize that all proofs and computations can be performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components, and we provide a picture of that for small genera.