# 2018年度 京都力学系セミナー

## 2018 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)

• 2月15日（金） (確率論セミナーと合同) 10時30分より 3号館127大会議室にて
Gérard Ben Arous 氏 (New York University)
Chaos for spherical Spin Glasses
Abstract:
For spherical Spin Glasses at low temperature, the Gibbs measure can be “chaotic", i.e. depend very strongly on temperature.
In a recent joint work with E. Subag and O. Zeitouni, we give a very detailed description of the Gibbs measure for mixed spherical Spin Glasses at low temperature, in the so-called 1RSB phase (for one step Replica-Symmetry-Breaking phase). This detailed geometric description of the Gibbs measure implies the existence of chaos, and much more. This fact has to be contrasted with the case of pure spherical models, where chaos does not exist, as proved by E. Subag.

• 1月25日（金）
Artem Raibekas 氏, Pablo G Barrientos 氏 (UFF, Brazil)
Robust nongeneric unfoldings of tangencies of large codimension
Abstract:

In the first talk we will describe the notions of tangencies (homoclinic and heterodimensional), codimension, and blenders. We will explain how to construct the tangencies in a robust manner using blenders but for the dynamics induced in the tangent bundle.

The second talk will put the above objects in the context of parametric families and non-generic unfoldings. That is, tangencies that unfold ''slowly'' with respect to the parameter. We will discuss how it is possible to make these unfoldings robust using again the mechanism of blenders.

• 1月18日（金）
Viktoria Vedyushkina 氏 (Lomonosov Moscow State University)
Fomenko-Zieshang invariants help to model complex integrable Hamiltonian systems by integrable billiards.
Abstract:

The first part of the talk will be devoted to bases, main ideas and constructions of Fomenko-Zieschang theory of invariants of integrable Hamiltonian systems. In fact, such graph invariants help effectively describe the topology of phase space foliated on the closures of solutions of integrable systems. This invariant ia complete, so two such systems have fiber-wise diffeomorphic foliations if and only if their invariants coincide. Some their extension also classifies structures of trajectories of such systems in the sense of topological (orbital) equivalence.

In the second part the piece-wise smooth case will be considered. Class of integrable billiards in flat domains was generalized in recent time by gluing elementary domains by some of their common boundaries.

Calculation of their Fomenko-Zieschang invariants shows that smooth integrable systems in mechanics and geodesic flows on orientable 2-surfaces often have the same structure of closures of trajectories. Even if billiard domain is a some complex, it is more simple to see interesting effects (bifurcation of Liouville tori, periodic critical trajectories, their stability) than in integrable systems of higher degree. So, it allows to say that "visual" billiard system in a suitable domains gives us a chance to describe the behavior of a complicated integrable system.

• 12月26日（水） 10時より 6号館 809セミナー室にて
Jeroen Lamb 氏 (Imperial College London)
Classification of 1D random homeomorphisms up to topological conjugacy
Abstract:
We discuss the classification of random orientation-preserving homeomorphisms of the interval and circle with diffusive noise, up to topological conjugacy of the random dynamical systems generated by their i.i.d. iterations. This is joint work with Doan Thai Son (Hanoi) , Julian Newman (Lancaster) and Martin Rasmussen (Imperial).

• 12月21日（金）14:30より
奥山 裕介 氏 (京都工芸繊維大学)
Berkovich射影直線上の力学系的同変量とBerkovich双曲幾何
Abstract:
Rumelyは、非アルキメデス的体上定義された射影直線上の有理関数が 潜在的両還元を持つかどうかの判定アルゴリズムを与えた際に、 Berkovich射影直線上の新たな力学系的同変量をいくつか導入した。 本講演ではBerkovich双曲幾何およびBerkovich射影直線上の 調和解析を復習するとともに、それら力学系的同変量の Berkovich双曲幾何を用いた大域的表示を与えたい。

• 12月7日（金）
Davit Karagulyan 氏 (九州大学, JSPS fellow)
On the Möbius disjointness conjecture and a class of three-interval exchange maps
Abstract:
I will start by surveying on the Möbius disjointness conjecture of P.Sarnak, which was introduced in 2010 and has initiated many studies. Then, I will discuss a result of J. Bourgain, which establishes the conjecture for a class of three-interval exchange maps. I will present a result, where we estimate the measure of the parameter set in his result. As a consequence, we will show that it has positive, but not full Hausdorff dimension.

• 11月30日（金）
Boris Hasselblatt 氏 (Tufts University)
Centralizers of hyperbolic flows
Abstract:
It is well-known and elementary'' that Anosov flows have trivial centralizer (Ghys); we (Bakker, Fisher and Hasselblatt) prove this for transitive expansive flows. Our main result gives a residual set of non-Anosov $C^\infty$ Axiom A flows with no cycles such that diffeomorphisms commuting with them are time-$t$ maps of the flow. This requires a study of the Lie group of commuting diffeomorphisms using the flow dynamics on the invariant "foliations." The novelty is that we study flows and under weaker dynamical assumptions.

• 11月16日（金）
篠田 万穂 氏 (慶應義塾大学)
零温度極限を持たない局所定数関数の構成について
Abstract:
零温度極限問題では，平衡測度の定義に用いられるポテンシャルの寄与を逆温度 パラメータで制御し，逆温度パラメータを無限大 (温度を零)にした際の平衡測 度の挙動を調べる．零温度極限が存在するかというのは基本的な問題であり，１ 次元記号力学系においては，ポテンシャルとして局所定数関数を取れば，零温度 極限は存在することが知られている. 一方，３次元以上の記号力学系では零温度 極限が存在しない局所定数関数がChazottesとHochmanにより構成されている．本 講演では，零温度極限が存在しない２次元記号力学系上の局所定数関数の構成に ついて述べる．高次元の有限型サブシフトのクラスが多様なサブシフトを含むと いうことが一次元と高次元の場合の本質的な違いを生む．本研究はJean-René Chazottes 氏(Ecole Polytechnique) との共同研究である．

• 11月2日（金） (関西確率論セミナーと共催) 3号 館108号室にて
盛田 健彦 氏 (大阪大学)
Random iteration of direct products of nonsingular transformations
Abstract:
For a measure-preserving transformation, its direct product is naturally defined and it is used to study ergodic-theoretic properties of the original transformation. For example, a measure-preserving transformation is weak-mixing if and only if its direct product is ergodic. In the case of a random dynamical system given by random iteration of a family of nonsingular transformations, we can consider the random iteration of the direct products of transformations in the family. I would like to talk about some ergodic-theoretic properties of such a random iteration of direct products and their application to the study of sample-wise ergodic behavior of the original random dynamical system.

• 10月26日（金）
David Martí-Pete 氏 (京都大学)
Wandering domains for entire functions of finite order in the class $\mathcal B$
Abstract:
Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces a function of infinite order. We construct the first examples of functions in the class $\mathcal B$ of finite order with wandering domains. In Bishop's example, as well as in our construction, the wandering domains are of oscillating type, that is, with an unbounded non-escaping orbit. To build such function, we use quasiconformal interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and thus, since functions in the class $\mathcal B$ have order at least $1/2$, we can achieve the smallest possible order. This is a joint work with Mitsuhiro Shishikura.

• 10月12日（金）(関西確率論セミナーと共催)
世良 透 氏 (京都大学)
Functional limit theorem for sojourns near indifferent fixed points
Abstract:
Interval maps with indifferent fixed points are typical examples of infinite ergodic transformations. They have been studied as models of intermittent phenomena, such as intermittent transitions to turbulence in convective fluids. In this talk, we will present a functional limit theorem for their occupation measures, where the limit processes are concentrated to indifferent fixed points.

• 10月5日（金）14:30より
Differential Galois Theory and Integrable Systems: a Survey
Abstract:
We will survey some of the last twenty years results on the applications of the Differential Galois theory to Integrable Systems. In particular, we will talk about three applications of the Differential Galois Theory:
1) Non-integrability results of finite dimensional dynamical systems by means of the variational equations.
2) Study of the integrability of a stochastic birth-dead model of dynamics of populations.
3) Study of spectral problems, most of them related to solitonic type partial differential equations, like Korteweg-de Vries (KdV) equation.

• 9月10日（月） 16時より
On the rate of convergence of semigroups of holomorphic self-maps at the Denjoy-Wolff point
Abstract:
Let $(\varphi_t)$ be a semigroup of holomorphic self-maps of the unit disc $\mathbb{D}$ with Denjoy-Wolff point $\tau\in \partial\mathbb{D}$. We study the rate of convergence of the trajectories of the semigroup to $\tau$, that is, given $z\in \overline{\mathbb{D}}$, we discuss the behavior of $|\varphi_{t}(z)-\tau|$ as $t$ goes to $+\infty$. We also make a brief survey of the role of those semigroups in Loewner theory.

• 7月20日（金）
Serge Troubezkoy 氏 (Aix-Marseille Université)
On the Ehrenfest wind-tree model
Abstract:
I will report on joint work with Alba Malaga Sabogal. In 1912 Paul et Tatyana Ehrenfest proposed the wind-tree model in order to interpret the ergodic hypothesis of Boltzmann. In the Ehrenfest wind-tree model, a point particle (the “wind”) moves freely on the plane and collides with the usual law of geometric optics with irregularly placed identical square scatterers (the “trees”). We show that for generic configurations (in the sense of Baire) the wind-tree model has very nice dynamics in a.e. direction: minimality, ergodicity, and infinite ergodic index.

• 6月29日（金）
Gabriel Vigny 氏 (Université de Picardie Jules Verne)
The bifurcation measure has maximal entropy
Abstract:
Consider a holomorphic family of rational maps of the Riemann sphere of degree d>1. We define a natural notion of entropy of bifurcation, mimickingthe classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy (joint work with De Thélin and Gauthier).

• 6月22日（金）
佐野 薫 氏 (京都大学)
semi-abel多様体の自己射の擬周期点について
Abstract:

有理数体などの数論的な体の上で定義された代数多様体の有理点に対し、高さと呼ばれる重要な量がある。 代数多様体の自己射が与えられたとき、その反復合成により高さがどのように増加するかを調べることは、Diophantus幾何の分野で応用の知られた重要な研究である。川口-Silvermanは点の軌道に対して算術次数という増大度を測る量を定義し、軌道がZariski-稠密であれば算術次数は（第一）力学系次数に一致すると予想した。

今回、川口-Silvermanの予想をsemi-abel多様体の自己射に対して示すことに成功し、その試みの中で、semi-abel多様体の自己射がある条件を満たすときには、ある有理点bが存在し、aが擬周期点であることとa-bがねじれ点になることが同値であることを証明した。

本講演では高さ関数の導入から始め、川口-Silvermanの予想を例を交えて紹介 し、semi-abel多様体に対する前述の主張の証明の概要を準同型の場合に述べる。時間が許せば同じ状況での川口-Silverman予想の証明の概要を述べる。

本研究は東京大学の松澤陽介氏との共同研究である。

• 6月15日（金）
中川 勝國 氏 (広島大学)
記号力学系におけるエントロピースペクトルの剛性問題
Abstract:
不変測度のKolmogorov-Sinaiエントロピーは、両側シフトの記号力学系では測度論的同型の完全不変量である(Ornstein)が，片側シフトにおいてはそうではない． 本講演では、不変測度のエントロピースペクトルと呼ばれる関数が完全不変量になり得るかという問題を調べ，得られた結果をいくつか紹介する． Kolmogorov-Sinaiエントロピーはエントロピースペクトルの特殊値として実現されるので、この問題を考えることは自然である．

• 6月8日（金）
Eric Bedford 氏 (Stony Brook University)
The shape of the Julia set of the Henon map
Abstract:
We will show that for a complex Henon map the Julia set has a certain amount of complex structure. However, it is never C^1 smooth and is never real analytic (even with singularities).

• 5月25日（金）
清水 雄貴 氏 (京都大学)
曲面上の点渦力学系
Abstract:

曲面上の流体力学において，曲面の形状の観点からの流体運動の定性的理解が求 められている．一般に流体方程式は無限次元であるものの，曲面上の非圧縮非粘 性流体で，渦度がデルタ関数の線形結合で表される点渦力学系と呼ばれる対象は 有限次元ハミルトン系として表される． 点渦力学系の時間発展方程式を与える に際し，流体力学系Green関数と呼ばれるLaplace-Beltrami作用素の基本解は中 心的役割を果たす．そのため点渦力学系のより詳細な力学系的性質を得る上で， 流体力学系Green関数を計算可能かつ解析的な表示を得ることが不可避となる．

本講演では，はじめに平面や球面，標準トーラス上の点渦力学系を例に，流れ場 の形状の違いが流れにどのような変化をもたらすかを紹介した後，それらの例を 包括する，非自明なKillingベクトル場が存在する曲面上の流体力学的Green関数 の計算可能な解析的表示の構成を述べる．なお標準トーラス上の点渦力学系に関 する結果は坂上貴之教授（京都大学）との共同研究に基づく．

• 5月18日（金）
篠原 克寿 氏 (一橋大学)
Generic super-exponential growth of number of periodic points for smooth partially hyperbolic diffeomorphisms
Abstract:

一様双曲型力学系の周期点の個数は周期に関して高々指数関数的に増大する．一 様双曲性が崩れた力学系ではどうなるかを考えるのは自然な問題である．本セミ ナーでは $C^\infty$ の部分双曲型力学系で $C^\infty$ 位相で通有的（generic） に超指数増大度を持つようなものが存在することを説明する．以下の技術的な点 を中心に説明をする予定である．

1. 一般の部分双曲型力学系では center leaf の間のホロノミーの正則性が低い （$C^1$が期待できない）．異なる leaf の力学系の情報をどう比較するか．
2. 周期点を作る際に heterodimensional cycle を足掛かりにして構成を行う． これを作るための常套手段である connecting lemma は一般には $C^1$ 位相の下 でしか使えない．この困難をどう解決するか．

本研究は浅岡正幸氏（京都大学），Dmitry Turaev氏（Imperial College London） らとの共同研究である．

• 4月20日（金）
高橋 博樹 氏 (慶応大学)
Large deviation principles for countable Markov shifts
Abstract:
For a finitely primitive topological Markov shift on a countably infinite number of alphabets we establish the large deviation principle. More precisely, we assume the existence of a Gibbs measure for a potential $\phi$ in the sense of Bowen, and prove the level-2 Large Deviation Principles for the distribution of Birkhoff averages under the Gibbs measure, as well as that of weighted periodic points and iterated pre-images. The rate function is in common, written with the pressure and the free energy associated with the potential $\phi$. The Gibbs measure is not assumed to be an equilibrium state for the potential $\phi$, nor is assumed the existence of an equilibrium state. We provide a sufficient condition for minimizers of the rate function to be equilibrium states.

• 4月13日（金）
Gal Binyamini 氏 (Weizmann Institute of Science)
Complex cells and the Yomdin-Gromov theorem
Abstract:
The Yomdin-Gromov algebraic lemma states that a semialgebraic subset of the unit cube can be parameterized by finitely many C^r-smooth charts of unit C^r norms, and that the number of these charts can be chosen to depend only on the degrees of the polynomials defining the set. This result is the main tool in Yomdin's proof of Shub's entropy conjecture for C^\infty-smooth maps. It has also been used more recently in the study of rational points on algebraic and transcendental varieties. Getting sharp control over the number of charts is the key to open problems in both the dynamical and diophantine directions. I will discuss a new complex analytic approach to the algebraic lemma. First, I will show that the appearance of C^k (rather than holomorphic) charts is due to obstructions related to the hyperbolic metric on the complex disc. I will then introduce "complex cells" complexifying the cellular decompositions of semialgebraic geometry, and show that their hyperbolic structure gives rise to a rich geometric function theory. Finally I will describe how using this function theory one can prove an essentially sharp version of the algebraic lemma. In particular this resolves a conjecture of Yomdin from 1991 on the topological entropy of real-analytic maps. The results are joint work with D. Novikov.

• 4月2日（月） (いつもと曜日が違います)
Dmitry Turaev 氏 (Imperial College London)
Exponential energy growth due to slow parameter oscillations in quantum mechanical systems
Abstract:
We use a classical adiabatic theorem to investigate the behavior of the Schroedinger equation in the domain whose shape varies slowly with time for the case where the domain is divided into two disjoint components during a part of the period of the slow oscillation. We show that this process is characterized by a periodic emergence and destruction of a certain additional quantum number and leads to an exponential growth of energy of the quantum mechanical system in question. In special cases, the process can also lead to a long period of cooling that precedes the acceleration, and to the desertion of the states with a particular value of the quantum number.

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