Eiko Kin

A family of pseudo-Anosov braids with small dilatation

1mCn : Matsuoka’s example [22] appears as 1;1 , and Ko, Los and Song’s example [18] as 2;1 . For any m;n 1, m;n is pseudo-Anosov (Theorem 3.9). The dilatations of m;m coincide with those found by Brinkmann [7] (see also Section 4.2), who also shows that the dilatations arising in this family can be made arbitrarily close to 1. It turns out that one may find smaller dilatations by passing a strand of m;n once around the remaining strands. As a particular example, we consider the braids m;n defined by taking the rightmost-strand of m;n and passing it counter-clockwise once around the remaining strands. Figure 1 gives an illustration of m;n and m;n . The braid 1;3 is conjugate to Ham and Song’s braid 1 2 3 4 1 2 . Forjm nj 1, we show that m;n is periodic or reducible. Otherwise m;n is pseudo-Anosov with

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.

Efficient topological chaos embedded in the blinking vortex system

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.

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

THE THIRD POWER OF THE SMALE HORSESHOE INDUCES ALL LINK TYPES

Let φ:D2→D2 be an orientation preserving homeomorphism of the disk into itself, and Φ= {φt}0≤t≤1 an isotopy with φ0=idD2 and φ1=φ. Then for a finite union of periodic orbits P of φ, the set is a link in D2×S1. We say that φ induces all link types (for Φ) if there exists a homeomorphism h of D2×S1 into a standardly embedded solid torus in the 3-sphere S3 such that any link L in S3 can be realized by a finite union of periodic orbits PL of φ so that L and are equivalent. We will show that the Smale horseshoe and its second power do not induce all link types, but its third power does induce all link types.

A SUSPENSION OF AN ORIENTATION PRESERVING DIFFEOMORPHISM OF D2 WITH A HYPERBOLIC FIXED POINT AND UNIVERSAL TEMPLATE

For an orientation preserving homeomorphism φ of the disk into itself, a suspension of a finite union of periodic orbits P of φ represents a link type in the 3-sphere S3. Let φ be a C1 diffeomorphism, and p a hyperbolic fixed point of φ with a homoclinic point. If all the homoclinic points for p are transeverse, then for infinitely many n>0, φn induces all link types, that is, for each link type L in S3, there exists a finite union of periodic orbits of φn such that a suspension of of φn represents L.