Yohei Komori

Landing property of stretching rays for real cubic polynomials

The landing property of the stretching rays in the parameter space of bimodal real cubic polynomials is completely determined. Define the Böttcher vector by the difference of escaping two critical points in the logarithmic Böttcher coordinate. It is a stretching invariant in the real shift locus. We show that stretching rays with non-integral Böttcher vectors have non-trivial accumulation sets on the locus where a parabolic fixed point with multiplier one exists.

Drawing bers embeddings of the teichmüller space of once-punctured tori

We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmüller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. For each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide whether the representation is discrete by applying Jørgensens theory on the quasi-Fuchsian space of once-punctured tori.

Bers embedding of the Teichmüller space of a once-punctured torus

In this note, we present a method of computing monodromies of projective structures on a once-punctured torus. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmüller space. As a by-product, the value of the accessory parameter of a four-times punctured sphere will be calculated in a numerical way as well as the generators of a Fuchsian group uniformizing it. Finally, we observe the relation between the Schwarzian differential equation and Heun’s differential equation in this special case.

Linear slices of the quasi-Fuchsian space of punctured tori

After fixing a marking (V,W ) of a quasi-Fuchsian punctured torus group G, the complex length λV and the complex twist τV,W parameters define a holomorphic embedding of the quasi-Fuchsian space QF of punctured tori into C2. It is called the complex Fenchel-Nielsen coordinates of QF . For c ∈ C, let Qγ,c be the affine subspace of C2 defined by the linear equation λV = c. Then we can consider the linear slice Lc of QF by QF ∩ Qγ,c which is a holomorphic slice of QF . For any positive real value c, Lc always contains the so-called Bers-Maskit slice BMγ,c defined in [Topology 43 (2004), no. 2, 447–491]. In this paper we show that if c is sufficiently small, then Lc coincides with BMγ,c whereas Lc has other components besides BMγ,c when c is sufficiently large. We also observe the scaling property of Lc.

An explicit counterexample to the equivariant K = 2 conjecture

We construct an explicit example of a geometrically finite Kleinian group G with invariant component Ω in the Riemann sphere Ĉ such that any quasiconformal map from Ω to the boundary of the convex hull of Ĉ − Ω in H3 which extends to the identity map on their common boundary in Ĉ, and which is equivariant under the group of Möbius transformations preserving Ω, must have maximal dilatation K > 2.002.

A Note on a Paper of Sasaki

In his paper [10], Sasaki studied the holomorphic slice S of the space of punctured torus groups determined by the trace equation xy = 2z. He found a simply connected domain E contained in S by using his system of inequalities which characterizes some quasifuchsian punctured torus groups (c.f. [9]). Moreover decomposing the boundary of E into 3 pieces ∂E = el U e2 U e3 he showed that e1 U e2 is contained in S and e3 (consisting of two points) is in the boundary ∂S. In this paper we consider the slice S itself more precisely.