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The minimum of the dilation factor for given genus is known for geni 1 and 2 only (the techniques behind could be extended to genus 3 but not farther). One also knows that $c_1/g\leq \log \delta(g)\leq c_2/g$.
Theorem (Boissy-Lanneau) Let $\Phi$ be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with $2g+2$ fixed points). Assume that $\Phi$ has a unique singularity. Then its dilation is strictly greater than $\sqrt{2}$ (but approach this value as $g\to\infty$).