# 2019年度 京都力学系セミナー

## 2019 Kyoto Dynamical Systems seminar

English

from 14:00, every Friday

Room 609 in Building no. 6 South Wing, at Facalty of Science, Kyoto University (Map)

 1月 24日 (金) 上原 崇人 氏 (岡山大学) (来年度に延期になりました)

• 1月17日（金）
沈 維孝 (Weixiao Shen) 氏 (復旦大学)
Primitive tuning via quasiconformal surgery
Abstract:
Using quasiconformal surgery, we prove that any primitive, postcritically-ﬁnite hyperbolic polynomial can be tuned with an arbitrary generalized polynomial, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves aﬃrmatively a conjecture by Inou and Jiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable. This is a joint work with Wang Yimin.

• 12月13日（金）14時30分から
Jan Kiwi 氏 (Pontificia Universidad Católica de Chile)
Indeterminacy Loci of Iterate Maps in Moduli Space
Abstract:
We study the action of the iteration maps on moduli spaces of complex rational maps. The tools employed emerge from considering dynamical systems acting on the Berkovich projective line over an appropriate non-Archimedean field.

The moduli space $\operatorname{rat}_d$ of rational maps in one complex variable of degree $d \ge 2$ has a natural compactification by a projective variety $\overline{\operatorname{rat}}_d$ provided by geometric invariant theory. Given $n \ge 2$, the iteration map $\Phi_n : \operatorname{rat}_d \to \operatorname{rat}_{d^n}$, defined by $\Phi_n: [f] \mapsto [f^n]$, extends to a rational map $\Phi_n: \overline{\operatorname{rat}}_d \dashrightarrow\overline{\operatorname{rat}}_{d^n}$. We characterize the elements of $\overline{\operatorname{rat}}_d$ which lie in the indeterminacy locus of $\Phi_n$. This is a joint work with Hongming Nie (Hebrew University of Jerusalem).

• 12月6日（金）
Fabrizio Bianchi 氏 (Université de Lille)
Stability and bifurcation in families of polynomial skew products
Abstract:
(Quadratic) polynomial skew-products are maps of the form $F (z, w) = (p(z), q(z, w))$, where $p$ and $q$ are polynomials of degree 2. These maps give the simplest non trivial examples of endomorphisms of P2(C). In this talk we investigate the natural parameter space of these maps, with emphasis on the stability-bifurcation dichotomy (that we will review in the beginning of the the seminar). In particular, we describe the geometry of the bifurcation current near infinity, and we give a partial classification of hyperbolic components. One of the tools we use is a generalisation to this setting of the one-dimensional equidistribution of some dynamically defined hypersurfaces of the parameter space towards the bifurcation current.

This is a joint work with Matthieu Astorg, Orléans.

• 11月22日（金） (応用数学セミナーと共催)
宮路 智行 氏 (京都大学)
A billiard problem arising from nonlinear and nonequilibrium systems
Abstract:
有界領域に閉じ込められたある種の自己駆動粒子は領域内部での直進と境界での反射を繰り返す．あたかもビリヤード球のようだが，境界に衝突せずに進行方向を変え，その反射規則は完全弾性反射ではないようである．そのため，数学的ビリヤード問題とは異なる様相が示される．このような運動は水面に浮かぶ円板状の樟脳や垂直に振動する液面を跳ねる液滴，平面上の反応拡散系やある種の非線形光共振器の数理モデルにおけるスポット解の運動など様々な系で観察されている．本講演では樟脳円板の運動を記述する数理モデル及びそこから中心多様体縮約によって導かれる常微分方程式モデルを通して，その運動の性質と矩形領域における軌道について議論する．

• 11月8日（金）15:30より
Walter Bergweiler 氏 (Christian–Albrechts–Universität zu Kiel)
Hyperbolic entire functions with bounded Fatou components
Abstract: (PDF)
The Eremenko-Lyubich class $B$ consists of all transcendental entire functions $f$ for which the set $\text{sing}(f^{-1})$ of critical and (finite) asymptotic values is bounded. A function $f\in B$ is called hyperbolic if every point of the closure of $\text{sing}(f^{-1})$ is contained in an attracting periodic basin. We show that if a hyperbolic map $f\in B$ has no asymptotic value and every Fatou component of $f$ contains at most finitely many critical points, then every Fatou component of $f$ is bounded. Moreover, the Fatou components are quasidisks in this case. If, in addition, there exists $N$ such that every Fatou component contains at most $N$ critical points, then the Julia set of $f$ is locally connected.

For hyperbolic maps in $B$ with only two critical values and no asymptotic value we find that either all Fatou components are unbounded, or all Fatou components are bounded quasidisks.

We illustrate the results by a number of examples. In particular, we show that there exists a hyperbolic entire function $f\in B$ with only two critical values and no asymptotic value for which all Fatou components are bounded quasidisks, but the Julia set is not locally connected.

The results are joint work with Núria Fagella and Lasse Rempe-Gillen.

• 11月1日（金）
Jan Kiwi 氏 (Pontificia Universidad Católica de Chile) (延期になりました)
宇敷 重廣 氏
Invariant curves in complex surface automorphisms
Abstract:
Automorphisms of complex surfaces can have various invariant curves. In this talk, we consider a family of rational surface automorphisms with an invariant caspidal cubic curve.

Such rational automorphism can have, at he same time, an invariant line, or an invariant quadratic curve, or a pair of lines intersecting at a point.

Dynamics in invariant curves are studied.

• 10月25日（金）
Matthieu Arfeux 氏 (Pontificia Universidad Católica de Valparaíso)
Trees and holomorphic dynamics
Abstract:
Some years ago Mistuhiro Shishikura showed how the use of some special trees and be useful to study existence of special holomorphic dynamical systems. Then, the same kind of trees have been introduced by DeMarco-McMullen to study some hyperbolic components of the space of polynomials dynamical systems. After the work of Jan Kiwi on Berkovich space, I introduced in my thesis a vocabulary to propose a definition of dynamical systems between trees of spheres in order to unify all of these points of view. With this vocabulary, Jan Kiwi's results have been reproved and I showed with Cui Guizhen how to improve our knowledge of special kind of behavior when taking sequences of rational maps diverging in the natural associated Moduli space.

• 10月18日（金）
James A. Yorke 氏 (University of Maryland) 3号館110号室にて (部屋が変更になりました)
Period Doubling Cascades - The big unsolved problem
Abstract:
Review of cascades results from the point of view of continuation theory and a description of the biggest problem that remains.

• 10月11日（金）
山中 祥五 氏 (京都大学)
2次元微分方程式系の可積分性とPoincaré-Dulac標準形への変換の収束性
Abstract:
平衡点近傍における解析手法として，ハミルトン系の場合にはBirkhoff標準形，一般的な場合にはPoincaré-Dulac標準形がある．一般に標準系への変換は収束するとは限らないベキ級数で与えられるが，Zung(2002)により，解析的に可積分であれば平衡点の近傍で収束する変換により標準化可能であることが示されている．この結果により，平衡点近傍における可積分性は収束する標準形への変換の存在と標準形自体の可積分性から調べることが可能と考えられる．このアイデアを適用し，ある2次元微分方程式系に対して，可積分であるための必要十分条件を与える．特に，標準形への変換の収束性を示すためにBorel変換を用いる．また，平衡点を持つ微分方程式に対して，共鳴次数1以下のPoincaré-Dulac標準形は必ず可積分であるという講演者の最近の結果も用いる．

• 10月4日（金）
高橋 博樹 氏 (慶応大学)
existence of large deviations rate function for any S-unimodal map
Abstract:
区間上のS-unimodal map, 特に2次写像が大偏差原理（LDP）を満たすことを示す。 以前に講演者らにより、高々有限回繰り込み可能で吸引周期点を持たない場合に、 最も深いrenormalization cycle上に制限した上でのLDPが（ほぼ）示されている。 今回の結果により、任意の2次写像について初期条件の制限なしにLDPが 示されたことになる。非常に複雑な分岐が起きているにも関わらず、 LDPが常に成立している点は興味深い。

本講演での内容はプレプリント arXiv:1908.07716 に収められている。

• 8月5日（月） 午前10時30分から 6号館809セミナー室にて
石川 勲 氏（理化学研究所/慶應義塾大学）
解析的な正定値関数から定まる再生核Hilbert空間における合成作用素 の有界性について
Abstract:
合成作用素(Koopman作用素）は複素解析的な文脈から古典的に調べられている対象であるが、近年では非線形な力学系モデルから生成される時系列データの解析や機械学習といった工学的な応用の文脈において高い注目を集めている。また、解析的な正定値関数から定まる再生核Hilbert空間は工学や統計において広く使われている対象である。ある写像の性質と写像の定める合成作用素の数学的な性質（有界性やコンパクト性など）との関係性は数学的に興味深い問題であると同時に、工学的な応用への理論保証を与えるためにも重要である。一方で、その関係性が知られているケースはあまり多くないのが現状である。本研究では、いくつか重要なケースで合成作用素の有界性ならば合成作用素を定める写像がcontractiveなAffine写像になるという事実を示したので、それについて証明の概要はアイデアについて概説したい。本研究は池田正弘氏（理研/慶應）、澤野嘉宏氏（首都大）との共同研究である。

• 7月26日（金）
Vadim Kaloshin 氏 (University of Maryland)
Birkhoff Conjecture for convex planar billiards
Abstract:
G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse). A famous conjecture by Birkhoff claims that ellipses are the only domains with this property. We show a local version of this conjecture - namely, that a small perturbation of an ellipse has this property only if it is itself an ellipse.

• 7月19日（金）
Andrzej J. Maciejewski 氏 (University of Zielona Góra)
Global residue theorem and integrability of homogeneous potentials
Abstract:
I will present an overview of my works connected with the integrability of natural Hamiltonian system with homogeneous potentials. An application of differential Galois methods for such system is effective as we known for them particular solutions. These solutions are defined by an algebraic set $\mathcal{D}$ in a complex projective space. It appears that residue of certain differential forms taken over points of $\mathcal{D}$ restricts the necessary conditions for the integrability deduced from the differential Galois theory.

• 6月21日（金）
色川 怜未 氏 (東京工業大学/理化学研究所)
Activity measures of dynamical systems over non-archimedean fields
Abstract:
Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line, an over non-archimedean ﬁelds, we study the stability (or passivity) of critical points of families of polynomials parametrized by an analytic curve. We construct the activity measure of a critical points of a family of polynomials, and study its property such as equidistribution, its relation to the Mandelbrot set.

• 6月14日（金）
金 英子 氏 (大阪大学)
Entropies of hyperbolic surface bundles over the circle as branched double covers of 3-manifolds
Abstract:
The branched virtual fibering theorem by Makoto Sakuma states that every closed orientable 3-manifold M with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle.
It is proved by Brooks and Montesinos that such surface bundle can be chosen to be hyperbolic. i.e, the monodromy map of such surface bundle can be chosen to be pseudo-Anosov. So it makes sense to talk about the topological entropies of hyperbolic surface bundles over the circle as branched double covers of M.
I discuss some properties of entropies of those hyperbolic surface bundles.
In joint work with Susumu Hirose, we prove that when M is the 3-sphere S^3, the minimal entropy over all hyperbolic, genus g surface bundles as branched double covers of S^3 behaves like 1/g.
If time permits, I will introduce some questions related to the branched virtual fibering theorem.

• 5月24日（金）
• Pieter Allaart 氏 (University of North Texas)
The pointwise Holder spectrum of self-affine functions
Abstract:
We study general self-affine functions on an interval, which include the Takagi function and Okamoto's functions. We show that the pointwise Holder spectrum of these functions can be completely determined. In most cases, the Holder spectrum is given by the multifractal formalism, but there is an important class of exceptions. In fact, it is possible to give exact (but complicated) expressions for the pointwise Holder exponent of any self-affine function at any point. The proofs of these results use a variety of techniques: Divided differenes, constrained optimization, and general Hausdorff measure estimates. This is joint work with S. Dubuc.
• 河邑 紀子 氏 (University of North Texas)
Revolving Fractals
Abstract:
Davis and Knuth in 1970 introduced the notion of revolving sequences, as representations of a Gaussian integer. Later, Mizutani and Ito pointed out a close relationship between a set of points determined by all revolving sequences and a self-similar set, which is called the Dragon from the viewpoint of symbolic dynamical systems. We will show how their result can be generalized by a completely different approach. The talk will be presented with a lot of pictures; accessible for graduate students. A few open problems will be introduced as well. This is a joint work with Drew Allen (UNT).

• 5月17日（金）
篠田 万穂 氏 (京都大学)
Intrinsic ergodicity for factors of ($-\beta$)-shift
Abstract:
We proved that every subshift factor of ($-\beta$) shifts is intrinsically ergodic, when $\beta$ is more than the golden ratio and the ($-\beta$)-expansion of $-1$ is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity beyond specification. We also prove that there exists a factor of $(-\beta)$-shift which is not intrinsically ergodic in the cases other than the above. This is a joint work with Kenichiro Yamamoto in Nagaoka University of Technology.

• 5月10日（金）
大林 一平 氏 (理化学研究所 AIP)
Persistent homology: Data analysis by algebraic topology
Abstract:

位相的データ解析というトポロジーの概念を活用したデータ解析分野がここ10〜20年発展しつつあり、特にパーシステントホモロジーという概念が重要となっている。歴史的にベッチ数を計算機で計算してデータ解析をしようというアイデアは古くからあったが、ノイズへの耐性の問題やトポロジカルな情報だけを取りだすのは情報量が少なすぎる、という問題があった。 パーシステントホモロジーはフィルトレーション上のホモロジーを考えることでこういった問題を解決した。

本講演では主に講演者の最近の2つの研究について紹介する
* パーシステントホモロジーの逆解析 (Volume-optimal cycles)
* パーシステントホモロジーと機械学習の組み合わせ
また、この2つの組み合わせがいかに強力か、といった話もする。 これらの話は両方とも数学(algebraic topology)と計算機科学(最適化や 機械学習など)の組み合わせによって実現されている。

• 4月26日（金）
Vassilis Rothos 氏 (Aristotle University of Thessaloniki)
Discrete and Continuous Nonlocal NLS Equation
Abstract:
In the first part, we study the existence and bifurcation results for quasi periodic traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions and with polynomial type potentials. We employ variational ana topological methods to prove the existence of traveling waves in nonlocal DNLS lattice. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential (nonlocal NLS). While in the case of cubic and quintic interactions of the same kind (e.g. both attractive or both repulsive), only a symmetry breaking bifurcation can be identified, a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions is a symmetry restoring'' bifurcation. Namely, in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the anti-symmetric state, there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the anti-symmetric parent one. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. The model is argued to be of physical relevance, especially so in the context of optical thermal media.

• 4月19日（金）
宇敷 重廣 氏
複素曲面の実断面上の力学系と複素サレム数
Abstract:
複素曲面のコクセター型の同型写像の力学系のコホモロジーへの作用の固有値と してサレム数が出現することが知られている。そうした複素力学系の、実軸に沿 う断面への制限は実曲面の力学系を誘導する。
この力学系によるホモロジーへの作用の固有値として、サレム数に類する、特殊 な代数的整数が出現する。
この複素数の固有値がレフシェツの不動点定理を通じて力学系のサドルの挙動と 結びついている。

〒606-8502 京都市左京区北白川追分町