Pseudo-Anosov Homeomorphisms on Translation Surfaces in Hyperelliptic Components Have Large Entropy

We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belongs to a hyperelliptic component is bounded from below uniformly by root 2 . This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surface of genus g tends to zero at rate 1/g (as g goes to infinity). We also show that our uniform lower bound root 2 is sharp. More precisely, the least dilatation of a pseudo-Anosov on a genus g > 1 translation surface in a hyperelliptic component belongs to the interval ]root 2, root 2 + 2(1-g)[. The proof uses the Rauzy-Veech induction.