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

This paper concerns the set <(M)over 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 is an element of Q. We prove that for each g (resp. g not equivalent to 0. mod 6)), the minimum among dilatations of elements ( resp. elements with orientable invariant foliations) of <(M)over defined on a closed surface Sigma(g) of genus g is achieved by the monodromy of some Sigma(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 equivalent to 6 (mod12) we find a new family of pseudo-Anosovs defined on Sigma(g) with orientable invariant foliations obtained from N (-6) or N (4) by Dehn filling both cusps. We prove that if delta(+)(g) is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on Sigma(g), then