Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior

This paper concerns the set M̂ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold N by Dehn filling three cusps with a mild restriction. Let N(r) be the manifold obtained from N by Dehn filling one cusp along the slope r ∈ ℚ. We prove that for each g (resp. g ≢(mod 6)), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of M̂ defined on a closed surface Σ_g of genus g is achieved by the monodromy of some Σ_g-bundle over the circle obtained from N(3/-2) or N(1/-2) by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case g ≡ (mod 12) we find a new family of pseudo-Anosovs defined on Σ_g with orientable invariant foliations obtained from N(-6) or N(4) by Dehn filling both cusps. We prove that if δ⁺_g is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on Σ_g, then where δ(D_n) is the minimal dilatation of pseudo-Anosovs on an n-punctured disk. We also study monodromies of fibrations on N(1). We prove that if δ_1,n is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with n punctures, then.