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Sep 17, 2013
09/13

by
Eiko Kin

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Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland's dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid...

Source: http://arxiv.org/abs/0711.4398v1

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Jun 30, 2018
06/18

by
Eiko Kin

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We study the magic manifold $N$ which is a hyperbolic and fibered $3$-manifold. We give an explicit construction of a fiber $F_a$ and its monodromy $:F_a \rightarrow F_a$ of the fibration associated to each fibered class $a$ of $N$. Let $\delta_g$ (resp. $\delta_g^+$) be the minimal dilatation of pseudo-Anosovs (resp. pseudo-Anosovs with orientable invariant foliations) defined on an orientable closed surface of genus $g$. As a consequence of our result, we obtain the first explicit...

Topics: Dynamical Systems, Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1412.7607

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Jun 29, 2018
06/18

by
Eiko Kin; Dale Rolfsen

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It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our work we find that of the two minimal volume hyperbolic 2-cusped orientable 3-manifolds, one has bi-orderable fundamental group whereas the other does not. We prove a similar result...

Topics: Group Theory, Geometric Topology, Mathematics

Source: http://arxiv.org/abs/1610.03241

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Sep 17, 2013
09/13

by
Eiko Kin; Mitsuhiko Takasawa

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The dilatation of a pseudo-Anosov braid is a conjugacy invariant. In this paper, we study the dilatation of a special family of pseudo-Anosov braids. We prove an inductive formula to compute their dilatation, a monotonicity and an asymptotic behavior of the dilatation for this family of braids. We also give an example of a family of pseudo-Anosov braids with arbitrarily small dilatation such that the mapping torus obtained from such braid has 2 cusps and has an arbitrarily large volume.

Source: http://arxiv.org/abs/0711.3009v1

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Sep 17, 2013
09/13

by
Eiko Kin; Mitsuhiko Takasawa

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Let $\delta_g$ be the minimal dilatation for pseudo-Anosovs on a closed surface $\Sigma_g$ of genus $g$ and let $\delta_g^+$ be the minimal dilatation for pseudo-Anosovs on $\Sigma_g$ with orientable invariant foliations. This paper concerns the pseudo-Anosovs which occur as the monodromies on closed fibers for Dehn fillings of $N(r)$ for each $r \in \{-3/2, -1/2, 2\}$ of the magic manifold $N$. The manifold $N(-3/2)$ is homeomorphic to the Whitehead sister link exterior. We consider the set...

Source: http://arxiv.org/abs/1003.0545v2

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Sep 20, 2013
09/13

by
Eiko Kin; Mitsuhiko Takasawa

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Let $\delta_{g,n}$ be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. Tsai proved that for any fixed $g \ge 2$, the logarithm of the minimal dilatation $\log \delta_{g,n}$ is on the order of $\frac{\log n}{n}$. The main result of this paper is that if $2g+1$ is relatively prime to $s$ or $s+1$ for each $0 \le s \le g$, then $$\limsup_{n \to \infty} \frac{n \log \delta_{g,n}}{\log n} \le 2.$$

Source: http://arxiv.org/abs/1205.2956v2

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Sep 22, 2013
09/13

by
Eiko Kin; Mitsuhiko Takasawa

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We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are genus 0 fiber surfaces, and describe their monodromies by braids. Among such homology classes whose representatives have n punctures, we decide which one realizes the...

Source: http://arxiv.org/abs/0812.4589v2

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Sep 21, 2013
09/13

by
Eiko Kin; Sadayoshi Kojima; Mitsuhiko Takasawa

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This paper concerns the set $\hat{\mathcal{M}}$ 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. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some...

Source: http://arxiv.org/abs/1104.3939v3

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Sep 22, 2013
09/13

by
Eiko Kin; Sadayoshi Kojima; Mitsuhiko Takasawa

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We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichm\"uller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and...

Source: http://arxiv.org/abs/0812.2941v1

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Sep 23, 2013
09/13

by
Eriko Hironaka; Eiko Kin

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This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3, 4 and 5 strands appear in this family. A pseudo-Anosov braid with 2g+1 strands determines a hyperelliptic mapping class with the same dilatation on a genus-g surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus-g surface grow asymptotically with the genus like 1/g, and gave explicit examples of mapping classes with dilatations...

Source: http://arxiv.org/abs/0904.0594v1

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Sep 19, 2013
09/13

by
Eriko Hironaka; Eiko Kin

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This paper concerns a family of pseudo-Anosov braids with dilatations arbitrarily close to one. The associated graph maps and train tracks have stable "star-like" shapes, and the characteristic polynomials of their transition matrices form Salem-Boyd sequences. These examples show that the logarithms of least dilatations of pseudo-Anosov braids on $2g+1$ strands are bounded above by $\log(2 + \sqrt{3})/g$. It follows that the asymptotic behavior of least dilatations of pseudo-Anosov,...

Source: http://arxiv.org/abs/math/0507012v1

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Jun 28, 2018
06/18

by
Susumu Hirose; Eiko Kin

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We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is...

Topics: Group Theory, Mathematics, Dynamical Systems, Geometric Topology

Source: http://arxiv.org/abs/1507.01671