2023年度 京都力学系セミナー

ハイブリッドで開催します (場合によってはオンラインで行う可能性もあります)．
対面での参加は基本的に講演者と学内者のみとします 5月からは学内者以外も対面参加可とします．

日 時:

金曜15時00分より

from Friday 15:00

場 所:

京都大学大学院理学研究科 3号館 108セミナー室 (地図）

Room 108 in Building no. 3 at Facalty of Science, Kyoto University (Map)
(only for hybrid seminars. In-person participation is restricted to members of Kyoto University and speakers)

• 過去のセミナー: 2022年度, 2021年度, 2020年度, 2019年度, 2018年度, 2017年度, 2016年度, 2015年度, 2014年度, 2013年度, 2012年度, 2011年度, 2010年度, 2009年度, 2008年度, 2007年度, 2006年度, 2005年度, 2004年度, 2003年度, 2002年度, 2001年度
• これまでの Kyoto Dynamics Days
• 宇敷 重廣氏の web ページは京都力学系セミナー のサポートの元、数学教室のサーバに置くことになりました。
  The web page of Shigehiro Ushiki is now hosted by Department of Mathematics, with the support of Kyoto Dynamical Systems Seminar.

講演のタイトルと概要（新しい順に並べてあります） Titles and Abstracts

• 3月15日（金）(対面のみでの開催です．Zoom配信はありません．)

  西口 純矢 氏（東北大学）

  Dynamical systems with multivalued motions: maximal time of existence and omega-limit sets

  Abstract:

  The theory of semi-dynamical systems has been extended to local semi-dynamical systems and multivalued semi-dynamical systems to study the dynamics of differential equations without uniqueness, differential inclusions, and control systems. However, a theory linking these dynamical systems is still missing. In this talk, we develop a dynamical systems theory of non-global multivalued semiflows on topological spaces to capture what are brought by the combination of the locality with respect to time and the property that motions are multivalued. For this purpose, we introduce a notion of multivalued quasi-semiflows, which gives a minimal model of non-global multivalued semiflows and makes the research of this talk more than just a synthesis of local semi-dynamical systems and multivalued semi-dynamical systems. Among other things, we show that if the multivalued motion of a point $x$ clusters on a compact set, then the $\omega$-limit set of $x$ becomes nonempty. Furthermore, this non-emptiness implies the infiniteness of the maximal time of existence of the multivalued motion of $x$ under a suitable assumption of a multivalued quasi-semiflow. The results of this talk will give a foundation of the dynamical systems theory of non-global multivalued semiflows on topological spaces, which are motivated by the differential systems mentioned above.

  参考文献： J. Nishiguchi, Dynamical systems with multivalued motions: maximal time of existence and omega-limit sets, in preparation.

  
  
• 3月4日（月）

  William Mance 氏 (Adam Mickiewicz University)

  Borel complexity of sets of normal numbers via generic points in subshifts with specification

  Abstract:

  We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various generalisations: generalised Lüroth series expansions and $\beta$-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in $[0,1)$. Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a $\Pi^0_3$-complete set, meaning that it is a countable intersection of $F_\sigma$-sets, but it is not possible to write it as a countable union of $G_\delta$-sets). We also solve a problem of Sharkovsky--Sivak on the Borel complexity of the basin of statistical attraction. The crucial dynamical feature we need is a feeble form of specification. All expansions named above generate subshifts with this property. Hence the sets of normal numbers under consideration are $\Pi^0_3$-complete.

  
  
• 1月19日（金）

  Maxime Chatal 氏 (Université de Paris)

  Almost-reducibility of quasi-periodic linear systems.

  Abstract:

  Our concern is the study of linear equations with quasi-periodic coefficients, which appear in many concepts. As a typical example, the well known Schrödinger equations with quasi-periodic potentials can be rewritten as quasi-periodic cocycles. The most common way to understand the behavior of such systems is by studying their stability, which we call reducibility. I will first give an introduction to the concepts of quasi-periodic cocycles, reducibility and almost-reducibility. In a second part, we will discuss strategy to obtain perturbative almost-reducibility.

  
  
• 1月12日（金）

  地引 知栄 氏 (東京工業大学)

  Fuchsian群上の孤立順序の構成

  Abstract:

  群上のleft-circular orderとは左からの群作用で不変な巡回順序のことである。orderの研究は基本群やブレイド群を通じて、種々の応用が知られている。ここで異なるorderを比較する際は、isolated orderというものが鍵を握ることが知られている。そこで、一次元力学系の側面からisolated orderをいくつかのFuchsian群上に構成する方法を解説する。なお、丸山氏との共同研究となる。

  
  
• 12月1日（金）

  色川 怜未 氏 (NTT基礎数学研究センタ)

  Non-archimedean and hybrid dynamics of Héenon maps

  Abstract:

  To study of the meromorphic degeneration of dynamics, the theory of hybrid spaces, established by Favre, is known to be a strong tool. In this talk, we apply this theory to the dynamics of Hénon maps; for a famiy of Hénon maps$\{H_t\}_t$ parametrized by a unit punctured disk meromorphically degenerating at the origin, we show that as $t\to 0$, the family of the invariant measures $\{\mu_t\}$ “weakly converges” to the measure on the Berkovich affine plane which is naturally defined by the family $\{H_t\}_t$, in the sense of the theory of hybrid spaces.

  
  
• 11月24日（金）

  上原 崇人 氏 (岡山大学)

  射影曲面上の双有理写像による力学系スペクトルについて

  Abstract:

  射影曲面上の双有理写像に対して, 力学系次数とよばれる量が定義される. 力学系次数は写像の反復合成による力学系の複雑さを表す量であり, Pisot 数もしくはSalem 数とよばれる代数的数になることが知られている. 本講演では, 力学系次数全体の集合である力学系スペクトルに焦点をあて, 力学系スペクトルがあるワイル群のスペクトル半径と一致することを示す. また, 力学系スペクトルはThurston等により研究された3次元双曲多様体の体積全体と似た構造をもつことを解説する.

  
  
• 11月17日（金）京都大学応用数学セミナーと共催

  川原田 茜 氏 (京都教育大学)

  セル・オートマトンが生成するフラクタルの一変数関数による分類について

  Abstract:

  フラクタルを特徴付ける際には一般にフラクタル次元を用いるが、これだけでは分類しきれない場合もある。本講演では、離散数理モデルのセル・オートマトン（CA）によって生成したフラクタルを一変数関数によって特徴付ける方法を紹介する。CAはフラクタル生成器としても知られており、軌道が自己相似性を持つ例が多く存在する。CAによってフラクタルを生成する場合、有限時間ステップまでの生成過程の図形（プレフラクタル）を構成するセルの個数を具体的に数え上げることができるため、この数え上げの極限を取ることによって一変数関数を得る。これらの一変数関数は、フラクタルの特徴を継承しており、フラクタルの分類指標として用いることが期待される。さらにこれらの関数は、数学的にも興味深い病的関数となっている例もあるので、これらの例についても紹介したい。

  
  
• 11月10日（金）

  アリババイ 仁真 氏 (京都大学)

  重み付き位相エントロピーとハウスドルフ次元

  Abstract:

  重み付き位相エントロピーという不変量は, 力学系のファクターの連なりと各ファクター上の重みに対して定義される量であり, Bedford-McMullenカーペット等の次元公式から着想を得て導入された. 最近, この量の別定義を一般化し, 変分原理を介して二つの定義が一致することを示した. この結果を用いて,　自己アフィンスポンジのハウスドルフ次元が初等的に計算でき, さらにある種の高次元ソフィック集合の次元も決定できる. 講演では証明のアイディアや, ハウスドルフ次元との結びつきについて具体例をもとに紹介する. (論文arXiv:2307.16772に基づく.)

  
  
• 10月27日（金）

  高橋 博樹 氏 (慶應義塾大学)

  The dynamics of heterochaos baker maps and its ramifications

  Abstract:

  3次元以上の多様体上の微分同相写像による力学系において、 不安定次元の異なるサドル型周期点が同一のtransitive setにロバストに稠密に 共存する現象は70年代から知られており、今日では主にheterodimensional cycleと blenderによる理解が得られている。本講演では、この現象を示すminimal model として[1]で導入、考察されたheterochaos baker mapと、それから派生する力学系の エルゴード的性質（maximal entropy measureの非一意性 [2,4]、非エントロピー稠密性 [2,4]、 相関の指数的減衰 [3]）について最近得られた結果をまとめて報告する。

  [1] S. Saiki, H. Takahasi, J.A. Yorke: Piecewise linear maps with heterogeneous chaos. Nonlinearity 36 (2023) 1776-1788
  [2] H. Takahasi, K. Yamamoto: Heterochaos baker maps and the Dyck system: maximal entropy measures and a mechanism for the breakdown of entropy approachability. Proc. Amer. Math. Soc. accepted in May 2023
  [3] H. Takahasi: Exponential mixing for heterochaos baker maps and the Dyck system. arXiv:2307.08119
  [4] H. Takahasi: Robust breakdown of intrinsic ergodicity and entropy-density for robustly transitive sets of diffeomorphisms. in preparation

  
  
• 10月13日（金）

  宇敷 重廣 氏

  Attracting Herman rings for rational maps on P^2

  Abstract:

  Some attracting invariant sets are observed numerically for dynamical systems defined by quadratic rational functions on P^2. These invariant objects seem to be isomorphic to Herman rings. They are found in families of quadratic rational maps modified from those birational maps inducing surface automorphisms. Possible bifurcations, which could generate such attractors, are indicated.

  
  
• 8月17日（木）午後1時半〜5時頃 (京都大学応用数学セミナーと共催)

  13:30-14:30 Allen Hart 氏 (Exeter University, UK)

  Uncertainty when forecasting with reservoir computers

  Abstract:

  Reservoir computers are a special type of recurrent neural network used for a variety of tasks including time series forecasting. The uncertainty of the forecast depends primarily on two quantities: the number of neurons $N$ comprising the reservoir computer and the number of training data $\ell$ comprising the time series. Under certain conditions, a central limit theorem applies causing the uncertainty on the forecast to converge to $0$ with order $1/\sqrt{N}$ and $1/\sqrt{\ell}$ in the number of neurons $N$ and data points $\ell$.
  The results hold for a broad class of time series, including ergodic dynamical systems arising from ODEs and stationary ergodic stochastic processes.

  
  

  15:00-16:00 Marian Mrozek 氏 (Jagiellonian University, Poland)

  Morse predecompositions in classical and combinatorial dynamics (based on work in progress with Michal Lipinski and Konstantin Mischaikow)

  Abstract:

  Morse decomposition separates gradient dynamics from recurrent dynamics by gathering all the recurrent behavior in isolated invariant sets known as Morse sets. However, the nature of recurrence inside Morse sets remains hidden. A more general concept of Morse predecomposition aims to reveal not only the separation but also the internal structure of Morse sets. I'll present it in the setting of combinatorial dynamical systems where the concept is easier to grasp, as well as in the setting of classical dynamics. I'll also present a potential of Morse predecompositions in combinatorial dynamics in providing new methods for computer assisted proofs of the existence of periodic orbits and chaotic invariant sets in classical dynamical systems.

  
  
• 7月28日（金）

  広中 えり子 氏 (Florida State University)

  On the connectivity of Per_n and its deformation space.

  Abstract:

  The space Per_n is the algebraic subspace of rat_2, defined as the space of quadratic rational maps modulo Mobius transformations with a marked critical cycle of period n. An open problem, due to J. Milnor (in the 1990's), is whether Per_n is irreducible for all integers n. In this talk I will discuss two approaches to Milnor's question. The first (joint with S. Koch) follows work of A, Epstein, where one studies at an analog of Per_n in the Teichmueller space of n+1 marked points on a sphere. The second is work in progress with C. Davis and A. Kapiamba, where Per_n is explored simultaneously as a parameter space with punctures and as an infinite collection of punctured algebraic curves in a complex affine plane.

  
  
• 7月21日（金）

  塚本 真輝 氏 (京都大学)

  ウエスト不等式のエントロピーと平均次元への応用．

  Abstract:

  ウエスト不等式というのは，等周不等式とボルスク・ウラムの定理を 融合したような定理であり，グロモフが2003年の論文で証明した． ウエスト不等式とその変種は様々な分野での応用が知られている． 例えば，凸幾何学，モース理論の一般化，組み合わせ論，複素解析などである． 最近，ウエスト不等式を位相力学系のエントロピーと平均次元の理論に応用することに 成功したので，それについて紹介する． この講演はRuxi Shiとの共著論文arXiv:2211.10158に基づく．

  
  
• 6月23日（金）

  Wei Hao Tey 氏 (東京大学)

  Bifurcation of minimal attractors of random dynamical systems with bounded noise

  Abstract:

  The importance of considering the presence of noise and uncertainty has become increasingly evident in real-world applications during the last few decades. With an assumption of bounded noise, the stationary distributions of the corresponding random dynamical systems are typically non-unique and supported on compact sets, allowing for a topological characterisation of the dynamical situation. The collection of trajectories with all possible noise realisations can be described at the topological level as a deterministic set-valued dynamical system. We are interested in the bifurcation of the stationary distribution of the random dynamical system, which turns out to be invariant sets of the set-valued system.
  Set-valued dynamical systems are notoriously challenging to analyse both theoretically and computationally, as they are defined on the set of compact subsets, which is infinite-dimensional and not amenable to analysis techniques typically used to understand bifurcation problems. We introduce a single-valued, finite-dimensional boundary map, inspired by the normal bundle of the boundary of invariant sets. We present an example of Hénon map with bounded noise where the bifurcation of the stationary distribution can be detected by traditional bifurcation of the boundary map. We also show some theoretical results on linear map and persistence of the normal bundle of invariant set’s boundary.

  
  
• 6月16日（金）

  鈴木 新太郎 氏 (東京学芸大学)

  Non-leading eigenvalues of the Perron-Frobenius operators for beta-maps

  Abstract:

  The beta-map for β > 1 is a simple piecewise linear expanding map on the unit interval and its ergodic properties can be investigated via its Perron-Frobenius operator, which is a bounded linear operator defined on the space of functions of bounded variation. On that space, the Perron-Frobenius operator is to be quasi-compact, i.e., any spectrum whose modulus is greater than 1/β is an isolated eigenvalue with finite multiplicity. In particular, it has 1 as its leading eigenvalue, although the less are known for the other isolated eigenvalues (non-leading eigenvalues), including their existence.
  In this talk, we see that the set of β’s such that the Perron-Frobenius operator corresponding to β has at least one non-leading eigenvalue is open and dense in (1, +∞) and that each non-leading eigenvalue is continuous but non-differentiable as a function of β defined on that set. Furthermore, we establish the Hölder exponent of that function for each non-leading eigenvalue.

  
  
• 6月9日（金）

  Sabyasachi Mukherjee 氏 (Tata Institute of Fundamental Research)

  Matings, holomorphic correspondences, and a Bers slice

  Abstract:

  There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: an algebraic correspondence framework due to Bullett-Penrose-Lomonaco and an orbit equivalence mating framework using Bowen-Series maps. The latter is analogous to the Douady-Hubbard theory for polynomial mating. We will discuss how these two frameworks can be unified and generalized. As a consequence, we will construct holomorphic correspondences that are matings of hyperbolic orbifold groups (including Hecke groups) with Blaschke products. Time permitting, we will introduce an analog of a Bers slice of the above orbifolds in the algebraic parameter space of correspondences.

  Based on joint work with Mahan Mj.

  
  
• 5月26日（金）

  Carsten Lunde Petersen 氏 (Roskilde University)

  Taming the elephants in the Mandelbrot set

  Abstract:

  We study classes of transcendental Horn and Adam maps, which contains the classical horn maps and their quotients coming from the dynamics on simple parabolic basins such as $z^2+1/4$ on the Cauliflower. We define and describe the structure of the dynamical and parameter “lagoa rays” and use these rays to design the Yoccoz puzzle machinery in this setting. As a first consequence, we obtain the precise asymptotics of secondary satellite copies of “principal elephants” near the cusp of the Mandelbrot set.

  
  
• 5月19日（金）

  本永 翔也 氏 (立命館大学)

  Constrained ergodic optimization for symbolic dynamics

  Abstract:

  Constrained ergodic optimization is a maximizing problem of the ergodic average for a given potential function in the set of all invariant probability measures with a given rotation vector. For symbolic dynamics, we prove that every relative maximizing measure for a generic potential has zero entropy, which is an analogical result of Morris’ theorem for the unconstrained case. In our proof, the density of periodic measures plays a key role. This is joint work with Mao Shinoda.

  
  
• 5月12日（金）

  梶原 唯加 氏 (京都大学)

  Variational structures for infinite transition orbits of monotone twist maps

  Abstract:

  This talk considers the chaotic dynamics and variational structures of area-preserving maps. To do so, we define a special class of area-preserving maps called monotone twist maps. Variational structures determined from monotone twist maps can be used to construct characteristic trajectories of these maps. Our goal is to define the variational structure such as giving `infinite transition orbits' through minimizing methods.

  
  
• 4月21日（金）

  大澤 知己 氏 (The University of Texas at Dallas)

  Relative Dynamics and Stability of Point Vortices on the Sphere

  Abstract:

  We present a Hamiltonian formulation of the dynamics of the ``shape'' of $N$ point vortices on the sphere: For example, if $N = 3$, it is the dynamics of the shape of the triangle formed by three point vortices, regardless of the position and orientation of the triangle on the sphere. Specifically, we first lift the dynamics of $N$ point vortices from the two-sphere $\mathbb{S}^{2}$ to $\mathbb{C}^{2}$, and then perform symplectic and Poisson reductions by symmetries to find a Poisson structure on the space of parameters for the shape of the point vortices. The resulting dynamics involves fewer shape variables than the previous work by Borisov and Pavlov on the same dynamics. As an application, we prove that the tetrahedron relative equilibria are stable when all of their circulations have the same sign, generalizing some existing results on tetrahedron relative equilibria of identical vortices on the sphere.

  
  
• 4月14日（金）

  稲生 啓行 (京都大学)

  A hole of the support of the bifurcation measure for the biquadratic family

  Abstract:

  双二次多項式のbifurcation measureは，ある種のMandelbrot集合の境界の一般化であり，Dujardin-Favreのlanding theoremを用いることで数値的に計算することができる．これをVRを用いて可視化し，インタラクティブに4次元的に回転させることで，「穴」が開いていることを発見した．この「穴」が実際に存在することの数値的な傍証と，それを精度保証計算を用いて厳密に証明するためのアイデアについて述べる．証明には放物型分岐とそこでのFatou座標の精密な評価が必要であり，それに関する現時点までの計算結果についても解説する．