We denote by δg (resp. δ + g ), the minimal dilatation for pseudo-Anosovs (resp. pseudo-Anosovs with orientable invariant foliations) on a closed surface of genus g. This paper concerns the pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the Whitehead sister link exterior W by Dehn filling two cusps, where the fillings are on the boundary slopes of fibers of W . We give upper bounds of δg for g ≡ 0, 1, 5, 6, 7, 9 (mod 10), δ + g for g ≡ 1, 5, 7, 9 (mod 10). Our bounds improve the previous one given by Hironaka. We note that the monodromies of fibrations on W were also studied by Aaber and Dunfield independently.

We consider a surface bundle over the circle, the so called magic manifold M . We determine homology classes whose minimal representatives are genus 0 fiber surfaces for M , and describe their monodromies by braids. Among those classes whose representatives have n punctures for each n, we decide which one realizes the minimal entropy. We show that for each n ≥ 9 (resp. n = 3, 4, 5, 7, 8), there exists a pseudo-Anosov homeomorphism Φ n : D n → D n with the smallest known entropy (resp. the smallest entropy) which occurs as the monodromy on an n-punctured disk fiber for the Dehn filling of M . A pseudo-Anosov homeomorphism Φ 6 : D 6 → D 6 with the smallest entropy occurs as the monodromy on a 6-punctured disk fiber for M .(Proposition 4.14). If gcd(m − 1, p) = 1, then the mapping torus T(Γ(T m,p )) is homeomorphic to M magic (Corollary 3.28). Otherwise Γ(T m,p ) is reducible. We setThe dilatation λ(b) ≈ 1.72208 equals the largest real root of f (5,3,2) (t) = t 6 − t 5 − 2t 3 − t + 1.(3b-ii) Γ(T 8k+7,2k+1 ) in case n = 8(k + 1) for k ≥ 1. The dilatation λ(T 8k+7,2k+1 ) equals the largest real root of f (4k+5,4k+1,0) (t) = t 8k+6 − 2(t 4k+1 + t 4k+5 ) + 1.

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We discuss a comparison of the entropy of pseudo-Anosov maps and the volume of their mapping tori. Recent study of the Weil-Petersson geometry of Teichmüller space tells us that the entropy and volume admit linear inequalities for both directions under some bounded geometry condition. Based on experiments, we present various observations on the relation between minimal entropies and volumes, and on bounding constants for the entropy over the volume from below. We also provide explicit bounding constants for a punctured torus case.