  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 Teichmueller geodesic flow restricted to the
hyperelliptic connected components, settling a question of Farb. We stress that
all proofs and computations are 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.
