Mitsuhiko Takasawa

Entropy versus Volume for Pseudo-Anosovs

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.

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

Conjugacy classes of the hyperelliptic mapping class group of genus 2 and 3

We present tables of conjugacy classes of the hyperelliptic mapping class group of genus 2 and 3, and some theorems on the Sp representation, the Jones representation, and Meyers function.