arXiv subjects

Tomoki Kawahira

Publications and source records attributed to Tomoki Kawahira.

14 recordsLinked to original sources

arXiv2026

Accessible hyperbolic components in anti-holomorphic dynamics

The tricorn, the connectedness locus of the anti-holomorphic quadratic family, is known to be non-locally connected. The boundary of every hyperbolic component of odd period contains arcs that are inaccessible from the complement of the tricorn. As the period increases, the decorations become more and more complicated, and it seems natural to think that every hyperbolic component of sufficiently large and odd period is inaccessible. Contrary to this expectation, we show that the tricorn contains infinitely many hyperbolic components that are accessible from the complement.

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arXiv2025

Julia sets appear quasiconformally in the Mandelbrot set, II: A parabolic proof

Following the ideas of A.~Douady, we give an alternative proof of the authors' result: for any boundary point $c_0$ of the Mandelbrot set $M$, we can find small quasiconformal copies of $M$ in $M$ that are encaged in nested quasiconformal copies of the totally disconnected Julia set of a parameter arbitrarily close to $c_0$.

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arXiv2024

Julia sets appear quasiconformally in the Mandelbrot set

In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". We can also find images of a Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. All the parameters belonging to these images are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set.

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arXiv2024

Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and investigation on the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia sets.

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arXiv2023

From Hyperbolic to Parabolic Parameters along Internal Rays

For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter $c$ converges to a parabolic parameter $\hat{c}$ radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on $\hat{c}$. We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at $\hat{c}$ with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors' paper ``From Cantor to semi-hyperbolic parameters along external rays" (Trans. Amer. Math. Soc. 372 (2019) pp. 7959--7992).

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arXiv2020

Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn

We present a simple proof of Tan's theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman's lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan's theorem.

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arXiv2019

From Cantor to Semi-hyperbolic Parameter along External Rays

For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\hat{c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt{|c-\hat{c}|})$ when $c$ belongs to a parameter ray that lands on $\hat{c}$. We also characterize the degeneration of the dynamics along the parameter ray.

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arXiv2018

Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set

For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that every point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the following derivative estimates of the motions near the boundary points $1/4$ and $-2$: for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt{1/4-c})$ when real $c\nearrow 1/4$; and is uniformly $O(1/\sqrt{-2-c})$ when real $c\nearrow -2$. These estimates of the derivative imply Hausdorff convergence of the Julia set $J(q_c)$ when $c$ approaches these boundary points. In particular, the Hausdorff distance between $J(q_c)$ with $0\le c<1/4$ and $J(q_{1/4})$ is exactly $\sqrt{1/4-c}$.

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arXiv2016

The Riemann hypothesis and holomorphic index in complex dynamics

We give an interpretation of the Riemann hypothesis in terms of complex and topological dynamics. For example, the Riemann hypothesis is affirmative and all zeros of the Riemann zeta function are simple if and only if a certain meromorphic function has no attracting fixed point. To obtain this, we use holomorphic index (residue fixed point index), which characterizes local properties of fixed points in complex dynamics.

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arXiv2014

Quatre applications du lemme de Zalcman à la dynamique complexe

We give four applications of Zalcman's lemma to the dynamics of rational maps on the Riemann sphere: a parameter analogue of a proof of the density of repelling cycles in the Julia sets;similarity between the Mandelbrot set and the Julia sets; a construction of the Lyubich-Minsky lamination and its variant; and a unified characterization of conical points by Lyubich-Minsky and those by Martin-Mayer.

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arXiv2011

On the natural extension of a map with a Siegel or Cremer point

In this note we show that the regular part of the natural extension (in the sense of Lyubich and Minsky) of quadratic map $f(z) = e^{2 πi θ}z + z^2$ with irrational $θ$ of bounded type has only parabolic leaves except the invariant lift of the Siegel disk. We also show that though the natural extension of a rational function with a Cremer fixed point has a continuum of irregular points, it can not supply enough singularity to apply the Gross star theorem to find hyperbolic leaves.

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arXiv2008

Tessellation and Lyubich-Minsky laminations associated with quadratic maps II: Topological structures of 3-laminations

According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient 3-laminations of Douady's rabbit, the Cauliflower, and $z \mapsto z^2$ are described. The descriptions employ a new method of tessellation inside the filled Julia set introduced in Part I that works like external rays outside the Julia set.

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arXiv2007

Topology of the regular part for infinitely renormalizable quadratic polynomials

In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topology is rigid modulo its combinatorics.

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arXiv2006

A proof of simultaneous linearization with a polylog estimate

We give an alternative proof of simultaneous linearization recently shown by T.Ueda, which connects the Schröder equation and the Abel equation analytically. Indeed, we generalize Ueda's original result so that we may apply it to the parabolic fixed points with multiple petals. As an application, we show a continuity result on linearizing coordinates in complex dynamics.

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