Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropy

TL;DR

This paper establishes a uniform lower bound of sqrt{2} for the dilatation of pseudo-Anosov homeomorphisms on hyperelliptic translation surfaces, contrasting with known asymptotic behaviors, and proves this bound is sharp.

Contribution

It provides the first uniform lower bound for dilatations in hyperelliptic components and demonstrates its sharpness, using Rauzy-Veech induction.

Findings

Lower bound of sqrt{2} for dilatations in hyperelliptic components

The minimal dilatation interval is between sqrt{2} and sqrt{2}+2^{1-g}

The bound is sharp for genus g > 1

Abstract

We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{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 sqrt{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 ]sqrt{2},sqrt{2}+2^{1-g}[. The proof uses the Rauzy-Veech induction.