Researcher basic information

• Doctor (Human and Environmental Studies), Kyoto University, May 2015

• https://researchmap.jp/yushi_nakano?lang=en

• https://jglobal.jst.go.jp/detail?JGLOBAL_ID=201501001162832699

• 201501001162832699

• Chaotic itinerary
• Additive noise
• Open systems
• Conformal measures
• Equilibrium states
• Thermodynamic Formalism
• Linear response theory
• Piecewise expanding maps
• Besov space
• Nagaev-Guivarc'h method
• Stein method
• Transient Chaos
• Almost sure invariant principle
• Gene co-expression network
• Neuroblastoma
• Teichmüller geodesic flow
• Zorich-Kontsevich cocycle
• Schottky group
• Stochastic Burgers cellular automaton
• Compound Poisson processes
• Wandering domains
• Partially hyperbolic systems
• SIR model
• Pitchfork bifurcation
• Saddle node bifurcation
• Intermittency
• Constrictivity
• Group action
• Arcsine law
• Infinite ergodic theory
• Palis conjecture
• Nonhyperbolic dynamical systems
• Foliation
• Entropy
• Microlocal analysis
• Lyapunov exponent
• Exel-Laca algebra
• Countable Markov shift
• Blender
• Homoclinic tangency
• Noise-indused phenomena
• Multiplicative noise
• Markov operator
• Limit theorems for dynamical systems
• Mixing
• Irregular set
• Stochastic stability
• Metastability
• Emergence
• Random dynamical system
• SRB measure
• Transfer operator
• Dynamical systems theory
• Ergodic theory

• Natural Science, Basic analysis, Dynamical systems theory, ergodic theory

• Bachelor's degree program, School of Science
• Master's degree program, Graduate School of Science
• Doctoral (PhD) degree program, Graduate School of Science

Research activity information

• Correction to: Abundance of Observable Lyapunov Irregular Sets
  Shin Kiriki; Xiaolong Li; Yushi Nakano; Teruhiko Soma
  Communications in Mathematical Physics, Jun. 2026
  Scientific journal
• Stationary Measures and Mean Flux Depending on Multiple Conserved Quantities in a Stochastic Cellular Automaton
  Kazushige Endo; Shinsuke Iwao; Hidetomo Nagai; Yushi Nakano
  27 May 2026
  We analyze a stochastic 5-neighbor cellular automaton with several conserved quantities, including the particle density. By examining the eigenvalue problem of the associated transition matrix, we derive an explicit formula for the stationary distribution on each irreducible component, in which the weight of each configuration is expressed in terms of the numbers of occurrences of two specific local patterns. This analysis further allows us to theoretically derive the dependence of the mean flux on the conserved quantities. In particular, we recover the mean flux formula in the deterministic case by taking the zero-noise limit of the system.
• A robust obstruction to full strong pluripotency for wild blender-horseshoes
  Shin Kiriki; Xiaolong Li; Yushi Nakano; Teruhiko Soma
  27 Apr. 2026
  Suppose that $M$ is a closed manifold of dimension greater than two and $r\geq 2$. We show that there exists a $C^r$-diffeomorphism $f:M\longrightarrow M$ with a wild affine blender-horseshoe $Λ_f$ which is $C^r$-robustly and strongly pluripotent for $Λ_f^{(\mathrm{mj})}$ but not for $Λ_f$, where $Λ_f^{(\mathrm{mj})}$ is the subset of $Λ_f$ consisting of elements with the majority condition. Thus, within the present affine blender-horseshoe family in [KNS], there is a robust obstruction to extending the strong pluripotency from $Λ_f^{(\mathrm{mj})}$ to the whole horseshoe.
• Functional correlation bound for random Lasota--Yorke maps with holes and its applications to conditional normal approximations
  Juho Leppänen; Yuto Nakajima; Yushi Nakano
  19 Apr. 2026
  This paper investigates the statistical properties of random open dynamical systems generated by families of Lasota--Yorke maps. Open systems, in which trajectories may escape through `holes', model transient phenomena and present additional difficulties for statistical analysis because the underlying ensemble loses mass over time. We show that the framework of functional correlation bounds (FCB), originally developed for closed systems, can also be adapted to this random open setting. The extension requires new ingredients based on Lasota--Yorke type inequalities in order to control the effect of escaping trajectories. We establish an FCB with exponential decay and combine it with the abstract normal-approximation results of \cite{LNN25,LS20} to obtain a conditional CLT with rates in Wasserstein distance and a conditional functional CLT with a rate in an integral distance over Barbour's class of smooth test functions. Additionally, we adapt Tikhomirov's method to obtain a bound in Kolmogorov distance for the conditional CLT.
• Non-existence of Lyapunov exponents in the Newhouse domain
  Shin Kiriki; Xiaolong Li; Yushi Nakano; Teruhiko Soma
  13 Apr. 2026
  We show that within the Newhouse domain of $C^r$ surface diffeomorphisms ($r \in [2,\infty )$), there exists a dense subset $\mathcal D$ such that for any $f \in \mathcal D$, Lyapunov exponents fail to exist for all points in some open set $U$ and all nonzero tangent vectors in some open cone $V \subset \mathbb{R}^2$. This demonstrates that the non-existence of Lyapunov exponents is a persistent phenomenon in the setting of robust homoclinic tangencies. The proof relies on constructing diffeomorphisms exhibiting specific oscillatory return times near a homoclinic tangency, incorporating techniques from Newhouse theory and recent results on Lyapunov irregularity, alongside several refinements and new arguments.
• Finitude of physical measures for Markovian random maps
  Pablo G. Barrientos; Dominique Malicet; Fumihiko Nakamura; Yushi Nakano; Hisayoshi Toyokawa
  17 Jul. 2025
  We study the finiteness of physical measures for skew-product transformations $F$ associated with discrete-time random dynamical systems driven by ergodic Markov chains. We develop a framework, using an independent and identically distributed (i.i.d.) representation of the Markov process, that facilitates transferring results from the well-studied Bernoulli (i.i.d.) setting to the Markovian context.
  Specifically, we establish conditions for the existence of finitely many ergodic, $F$-invariant measures, absolutely continuous with respect to a reference measure, such that their statistical basins of attraction for measurable bounded observables cover the phase space almost everywhere. Furthermore, we investigate a weaker notion, which demands finitely many physical measures (not necessarily absolutely continuous) whose weak$^*$ basins of attraction cover the phase space almost everywhere. We show that for random maps on compact metric spaces driven by Markov chains on finite state spaces, this property holds if the system is mostly contracting, i.e., if all the Markovian invariant measures have negative maximal Lyapunov exponents. This result is applied to random $C^1$ diffeomorphisms of the circle and the interval under conditions based on the absence of invariant probability measures or finite invariant sets, respectively. We also connect our result to the quasi-compactness of the Koopman operator on the space of Hölder continuous functions.
• Brownian approximation of dynamical systems by Stein's method
  Juho Leppänen; Yuto Nakajima; Yushi Nakano
  23 Jan. 2025
  We adapt Stein's method of diffusion approximations, developed by Barbour, to
  the study of chaotic dynamical systems. We establish an error bound in the
  functional central limit theorem with respect to an integral probability metric
  of smooth test functions under a functional correlation decay bound. For
  systems with a sufficiently fast polynomial rate of correlation decay, the
  error bound is of order $O(N^{-1/2})$, under an additional condition on the
  linear growth of variance. Applications include a family of interval maps with
  neutral fixed points and unbounded derivatives, and two-dimensional dispersing
  Sinai billiards.
• Fine construction of gene coexpression network analysis using GTOM and RECODE detected a critical module of neuroblastoma stages 4 and 4S
  Fumihiko Nakamura; Yushi Nakano; Shiro Yamada
  Hereditas, 161, 1, Dec. 2024, [Peer-reviewed]
  Scientific journal
• Length averages for codimension one foliations
  Masayuki Asaoka; Yushi Nakano; Paulo Varandas; Tomoo Yokoyama
  04 Nov. 2024
  In this paper we study geometrical and dynamical properties of codimension
  one foliations, by exploring a relation between length averages and ball
  averages of certain group actions. We introduce a new mechanism, which relies
  on the group structure itself, to obtain irregular behavior of ball averages
  for certain non-amenable group actions. Several geometric realization results
  show that any such groups can appear connected with the topology of leaves
  which are connected sums of plugs with a special geometry, namely nearly
  equidistant boundary components. This is used to produce the first examples of
  codimension one $\mathcal C^\infty$ regular foliations on a compact Riemannian
  manifold $M$ for which the length average of some continuous function does not
  exist on a non-empty open subset of $M$.
• Takens' Last Problem and strong pluripotency
  Shin Kiriki; Xiaolong Li; Yushi Nakano; Teruhiko Soma; Edson Vargas
  27 Apr. 2024
  We consider the concept of strong pluripotency of dynamical systems for a
  hyperbolic invariant set, as introduced in [KNS]. To the best of our knowledge,
  for the whole hyperbolic invariant set, the existence of robust strongly
  pluripotent dynamical systems has not been proven in previous studies. In fact,
  there is an example of strongly pluripotent dynamical systems in [CV01], but
  its robustness has not been proven. On the other hand, robust strongly
  pluripotent dynamical systems for some proper subsets of hyperbolic sets had
  been found in [KS17, KNS]. In this paper, we provide a combinatorial way to
  recognize strongly pluripotent diffeomorphisms in a Newhouse domain and prove
  that they are $C^r$-robust, $2\leq r< \infty$. More precisely, we prove that
  there is a 2-dimensional diffeomorphism with a wild Smale horseshoe which has a
  $C^r$ neighborhood $\mathcal{U}_0$ where all elements are strongly pluripotent
  for the whole Smale horseshoe. Moreover, it follows from the result that any
  property, such as having a non-trivial physical measure supported by the Smale
  horseshoe or having historic behavior, is $C^r$-persistent relative to a dense
  subset of $\mathcal{U}_0$.
• Pluripotency of wandering dynamics
  Shin Kiriki; Yushi Nakano; Teruhiko Soma
  30 Mar. 2024
  This paper proposes a new concept of pluripotency inspired by Colli-Vargas
  [Ergod. Theory Dyn. Syst., 21(6):1657-1681, 2001] and presents fundamental
  theorems for developing the theory. Pluripotency reprograms dynamics from a
  statistical or geometrical point of view. This means that the dynamics of
  various codes, including non-trivial Dirac physical measures or historic
  behavior, can be observably and stochastically realized by arbitrarily small
  perturbations. We first give a practical condition equivalent to a stronger
  version of pluripotency. Next, we show that the property of pluripotency is
  $C^{r} (2\leq r<\infty)$-robust. Precisely, there exists a $C^{r}$-open set of
  non-hyperbolic diffeomorphisms that have wild blender-horseshoes and are
  strongly pluripotent. It implies a new affirmative solution to Takens' last
  problem for $C^{r}$ diffeomorphisms of dimension $n\geq 3$.
• A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics
  Harry Crimmins; Yushi Nakano
  Ergodic Theory and Dynamical Systems, 44, 4, 20 Jun. 2023, [Peer-reviewed]
  English, Scientific journal
• Historic and physical wandering domains for wild blender-horseshoes
  Shin Kiriki; Yushi Nakano; Teruhiko Soma
  Nonlinearity, 36, 8, 4007, 4033, 16 Jun. 2023, [Peer-reviewed]
  English, Scientific journal
• Observable Lyapunov irregular sets for planar piecewise expanding maps
  Yushi Nakano; Teruhiko Soma; Kodai Yamamoto
  Discrete and Continuous Dynamical Systems, American Institute of Mathematical Sciences (AIMS), 01 Apr. 2023, [Peer-reviewed]
  Scientific journal, For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family
  $F_\sigma$ $(0<\sigma<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding
  maps such that each $F_\sigma$ has an observable (i.e. Lebesgue positive)
  Lyapunov irregular set. These maps are obtained by modifying the piecewise
  expanding map given in Tsujii (2000). In strong contrast to it, we also show
  that any Lyapunov irregular set of any 2-dimensional piecewise real analytic
  expanding map is not observable. This is based on the spectral analysis of
  piecewise expanding maps in Buzzi (2000).
• Arcsine law for random dynamics with a core
  Fumihiko Nakamura; Yushi Nakano; Hisayoshi Toyokawa; Kouji Yano
  Nonlinearity, 36, 3, 1491, 1509, 01 Mar. 2023, [Peer-reviewed]
  English, Scientific journal
• Quenched limit theorems for random U(1) extensions of expanding maps
  Yong Moo Chung; Yushi Nakano; Jens Wittsten
  DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 43, 1, 338, 377, 04 Jan. 2023, [Peer-reviewed]
  English, Scientific journal
• A study to identify spontaneous regression pathways by comparing gene co-expression graphs of neuroblastoma stage 4 and 4S
  山田思郎; 中村文彦; 中野雄史; 桐木紳; 井ノ上逸朗
  日本小児血液・がん学会雑誌(Web), 60, 4, 2023
• Topological entropy for countable Markov shifts and Exel--Laca algebras
  Yuta Michimoto; Yushi Nakano; Hisayoshi Toyokawa; Keisuke Yoshida
  30 Dec. 2022
  We show that the (Gurevich) topological entropy for the countable Markov
  shift associated with an infinite transition matrix $A$ coincides with the
  non-commutative topological entropy for the Exel--Laca algebra associated with
  $A$, under certain conditions on $A$. An important example satisfying the
  conditions is the renewal shift, which is not locally finite. We also pose
  interesting questions for future research on non-commutative topological
  entropy for non-locally finite transition matrices.
• Finitude of physical measures for random maps
  Pablo G. Barrientos; Fumihiko Nakamura; Yushi Nakano; Hisayoshi Toyokawa
  19 Sep. 2022
  For random compositions of independent and identically distributed measurable
  maps on a Polish space, we study the existence and finitude of absolutely
  continuous ergodic stationary probability measures (which are, in particular,
  physical measures) whose basins of attraction cover the whole space almost
  everywhere. We characterize and hierarchize such random maps in terms of their
  associated Markov operators, as well as show the difference between classes in
  the hierarchy by plenty of examples, including additive noise, multiplicative
  noise, and iterated function systems. We also provide sufficient practical
  conditions for a random map to belong to these classes. For instance, we
  establish that any continuous random map on a compact Riemannian manifold with
  absolutely continuous transition probability has finitely many physical
  measures whose basins of attraction cover Lebesgue almost all the manifold.
• Emergence via non-existence of averages
  Shin Kiriki; Yushi Nakano; Teruhiko Soma
  Advances in Mathematics, 400, 14 May 2022, [Peer-reviewed]
  English, Scientific journal
• Abundance of Observable Lyapunov Irregular Sets
  Shin Kiriki; Xiaolong Li; Yushi Nakano; Teruhiko Soma
  Communications in Mathematical Physics, 391, 3, 1241, 1269, May 2022, [Peer-reviewed]
  English, Scientific journal
• (ランダム)力学系におけるLyapunov非正則集合の観測可能性—ランダム力学系および多価写像力学系理論の総合的研究
  中野 雄史
  数理解析研究所講究録, 2217, 144, 152, Apr. 2022
  Japanese
• QUENCHED EXPONENTIAL MIXING FOR RANDOM EXPANDING SEMIFLOWS (Research on the Theory of Random Dynamical Systems and Fractal Geometry)
  Nakano, Yushi
  RIMS Kokyuroku, 2176, 50, 56, 京都大学数理解析研究所, Apr. 2021
  Japanese, Research institution
• Large intersection classes for pointwise emergence
  Nakano Yushi
  RIMS Kokyuroku, 2181, 2181, 103, 111, Research Institute for Mathematical Sciences, Kyoto University, Apr. 2021
  Japanese, Research institution, 非正則集合（エルゴード平均が存在しないような点からなる集合）の複雑さを定量的に測るものとして，近年，桐木紳氏，相馬輝彦氏，および報告者によって各点創発の概念が導入された．そこではフルシフトについて，高創発集合（各点創発が超多項式的である点からなる集合；非正則集合の部分集合となる）が位相的に大きい，つまり通有的であることが示された．本稿では，報告者が最近A.Zelerowicz氏と共同で得た次の結果について報告する：有限型部分シフトについて，高創発集合の位相的エントロピーは力学系の位相的エントロピーと一致し，高創発集合のHausdorff次元は相空間のHausdorff次元と一致し，任意のHolder連続なポテンシャルに関して高創発集合の位相的圧力は力学系の位相的圧力と一致する（つまり，高創発集合は熱力学形式的に大きい集合である）．これらはすべて，Caratheodory次元の交叉安定性に関するより一般的な結果の系として得られる．
• Lyapunov Exponents for Random Maps
  Fumihiko Nakamura; Yushi Nakano; Hisayoshi Toyokawa
  DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, 27, 12, 7657, 7669, 22 Mar. 2021, [Peer-reviewed]
  English, Scientific journal
• Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems
  Pablo G. Barrientos; Yushi Nakano; Artem Raibekas; Mario Roldan
  DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, 37, 2, 181, 210, 20 Mar. 2021, [Peer-reviewed]
  English, Scientific journal
• Mixing and observation for Markov operator cocycles
  Fumihiko Nakamura; Yushi Nakano; Hisayoshi Toyokawa
  NONLINEARITY, 35, 1, 66, 83, 28 Sep. 2020, [Peer-reviewed]
  English, Scientific journal
• Highly irregular orbits for subshifts of finite type: large intersections and emergence
  Yushi Nakano; Agnieszka Zelerowicz
  NONLINEARITY, 34, 11, 7609, 7632, 01 Sep. 2020, [Peer-reviewed]
  English, Scientific journal
• Historic behavior in nonhyperbolic homoclinic classes
  Pablo G. Barrientos; Shin Kiriki; Yushi Nakano; Artem Raibekas; Teruhiko Soma
  Proceedings of the American Mathematical Society, 148, 3, 1195, 1206, 2020, [Peer-reviewed]
  English, Scientific journal
• Irregular sets for piecewise monotonic maps
  Yushi Nakano; Kenichiro Yamamoto
  TOKYO JOURNAL OF MATHEMATICS, 44, 2, 495, 506, 27 Dec. 2019, [Peer-reviewed]
  English, Scientific journal
• Existence and Non-existence of Length Averages for Foliations
  Yushi Nakano; Tomoo Yokoyama
  Communications in Mathematical Physics, 372, 2, 367, 383, 01 Dec. 2019, [Peer-reviewed]
  English, Scientific journal
• Spectra of expanding maps on besov spaces
  Yushi Nakano; Shota Sakamoto
  Discrete and Continuous Dynamical Systems- Series A, 39, 4, 1779, 1797, Apr. 2019, [Peer-reviewed]
  English, Scientific journal
• Historic behaviour for nonautonomous contraction mappings
  Shin Kiriki; Yushi Nakano; Teruhiko Soma
  Nonlinearity, 32, 3, 1111, 1124, 13 Feb. 2019, [Peer-reviewed]
  English, Scientific journal
• Non-trivial wandering domains for heterodimensional cycles
  Shin Kiriki; Yushi Nakano; Teruhiko Soma
  Nonlinearity, 30, 8, 3255, 3270, 21 Jul. 2017, [Peer-reviewed]
  English, Scientific journal
• Historic behaviour for random expanding maps on the circle
  Yushi Nakano
  Tokyo Journal of Mathematics, 40, 1, 165, 184, Jun. 2017, [Peer-reviewed]
  English, Scientific journal
• Historic behaviour for random expanding maps on the circle (The Theory of Random Dynamical Systems and Its Applications)
  中野 雄史
  数理解析研究所講究録, 2028, 2028, 81, 90, 京都大学数理解析研究所, May 2017
  Japanese, Research institution, Takensは円周上の拡大写像について, 歴史的挙動を示すような初期値全体の集合が相空間上で残留的となることを示した. 本稿ではこの円周上の拡大写像の統計的性質が, 急冷型ランダム微小摂動によって保存されることを報告する. 証明はランダムなMarkov分割を構成することで得られるが, この分割の存在は(折畳み写像との位相共役に関する)ランダムな設定におけるShubの定理を示すことにより保証される. この副産物として, 円周上のランダム拡大写像の絶対連続でエルゴード的な不変確率測度に関する新しい公式を得る.
• Stochastic stability for fiber expanding maps via a perturbative spectral approach
  Yushi Nakano
  Stochastics and Dynamics, 16, 4, 01 Aug. 2016, [Peer-reviewed]
  English, Scientific journal
• The partial captivity condition for U(1) extensions of expanding maps on the circle
  Yushi Nakano; Masato Tsujii; Jens Wittsten
  Nonlinearity, 29, 7, 1917, 1925, 25 May 2016, [Peer-reviewed]
  English, Scientific journal
• ランダムに摂動されたトーラス上の部分拡大写像のスペクトル (ランダム力学系理論とその応用 : RIMS研究集会報告集)
  中野 雄史
  数理解析研究所講究録, 1942, 1942, 66, 77, 京都大学数理解析研究所, Apr. 2015
  Japanese, Research institution
• On the spectra of quenched random perturbations of partially expanding maps on the torus
  Yushi Nakano; Jens Wittsten
  Nonlinearity, 28, 4, 951, 1002, Apr. 2015, [Peer-reviewed]
  English, Scientific journal

• 微分積分学Ⅱ, 2024年, 学士課程, 全学教育
• 解析学Ｂ, 2024年, 学士課程, 理学部

• 準安定カオスの統計数理と現象制御への新視点
  戦略的な研究開発の推進 戦略的創造研究推進事業 さきがけ
  2025 - 2028
  中野 雄史
  観測時間内に限定した複雑系の長時間挙動を捉えるため、準安定カオスに対する独自の統計数理(観測時間依存の極限定理・熱力学形式等)を構築します。これをホモクリニック分岐等の汎用性の高い力学系へ適用し、その統計的性質の厳密な理解を通じて多様な応用への基盤を築きます。最終的に、本統計数理を複雑な化学反応経路の局所平衡解析などに適用し、現象の精緻な予測と効率的な制御を可能にする新たな視点を提示します。
  科学技術振興機構, 北海道大学, Principal investigator
• Study of physical measures for random non-hyperbolic dynamics
  Grants-in-Aid for Scientific Research
  01 Apr. 2023 - 31 Mar. 2027
  中野 雄史
  昨年度までに引き続き、ランダムな非双曲力学系における物理測度の性質解明を目指し、国内外の研究者との共同研究を積極的に推進した。これに関連して、以下のような成果を得た。
  (1) Araujoの結果を独立同分布ノイズからMarkov型ノイズへと拡張することに成功し、物理測度の存在と有限性に関する包括的な枠組みを構築した（中村文彦氏・豊川永喜氏（北見工大）、P. Barrientos氏（UFF）との共同研究）。これは当初目標の重要な達成点である。さらに、その過程で「Markov型ノイズへの移行原理」とも呼ぶべき高い一般性をもつ証明手法に到達し、これを利用することで、Barrientos-Malicetの独立同分布ノイズに関する概縮小写像系の物理測度の存在と有限性の結果を、Markov型ノイズへと拡張することにも成功した（前出3名およびD. Malicet氏（UGE）との共同研究）。
  (2) ノイズ下での極限定理（中心極限定理、大偏差原理、概不変原理、極値理論など）に関する研究を発展させ、とくに準安定性との関連に着目した「条件付き極限定理」の理論構築を進めた（S. Vaienti氏（CIRM）、G. Froyland氏・J. Atnip氏（UNSW）、C. Gonzalez-Tokman氏（Queensland）との共同研究）。これは決定論的力学系においても新規性のある成果であり、条件付き大数の強法則の定式化と証明という新しい局面を迎えている。
  (3) 汎関数相関減衰評価という最新のエルゴード理論の解析技術を、ランダムな開力学系へと拡張し、条件付き（汎関数）中心極限定理における収束レート評価への応用も実現した（J. Leppanen氏（東海大）、中島由人氏（同志社大）との共同研究）。
  これらの成果の一部は、国際誌に投稿中または投稿準備中である。
  Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tokai University, 23K03188
• Dynamical systems with observable Lyapunov irregular sets
  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
  01 Apr. 2022 - 31 Mar. 2026
  相馬 輝彦; 桐木 紳; 中野 雄史
  本研究の目的は，時間平均をとっても挙動が把握できないような非正則な力学系が豊富に存在することを，Lyapunov 指数を指標として明らかにすることにある．一昨年度は，共同研究者ある桐木紳氏（東海大学教授），中野雄史氏（北海道大学准教授），李曉龍氏（華中科技大学専任講師）および本研究代表者およびの4名で，Colli-Vargas モデルを改良することにより，観測可能な Lyapunov 非正則集合をもつ2次元力学系のモデルが構成できた．昨年度は，この結果の一般化を考えた．そのためには，Colli-Vargas モデルの摂動を考える必要があるが，そのような写像は馬蹄型不変集合上で，アファイン写像とはならないので，定理の証明には緻密な評価が必要であることが分かった．この問題は，本研究代表者，分担者にEdson Vargas 氏（ブラジル Instituto de Matematica e Estatistica 教授）を加えた4名による，2次元写像に関する強多様性(strong pluripotency)を研究の過程で完成した．この結果により，本来は本研究課題と独立の研究課題研究である「（強）多様性」という新しい力学系の概念が密接に関係してくることが明らかになった．2024年度は，観測可能な Lyapunov 非正則集合をもつ2次元力学系のモデルの一般化に成功した．現在この結果は，本研究代表者，分担者に李曉龍氏を加えた4名の共著論文として「Non-existence of Lyapunov exponents in the Newhouse domain」という題名で論文としてまとめているところである．この論文が完成したら，2025年度中に専門誌に投稿する予定である．
  Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tokyo Metropolitan University, 22K03342
• Existence and persistence of historic wandering domains for high-dimensional dynamical systems
  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
  01 Apr. 2021 - 31 Mar. 2026
  桐木 紳; 相馬 輝彦; 中野 雄史
  本研究課題の準備として、有用な例の作成をおこなってきた。まず概要を述べる。３次元アフィン写像でブレンダー馬蹄をもち、さらにそのブレンダー馬蹄のサドル周期点に関するホモクリニック接触をおこすような例を考え、その例の任意に小さなCr近傍に本研究課題で着目している性質であるヒストリック挙動やディラック物理測度を有することを調査し、証明することに成功した。この結果を論文にまとめ、arXivで公開した。またこの成果を国際研究集会の招待講演と国内の研究集会にて一般講演をおこなった。
  上のことを少し詳しく記述する。ブレンダーとは３次元以上の力学系の概念である。具体的には双曲的不変集合なのだが、その不変集合の不変多様体がある開領域で稠密になる。例えば２次元の馬蹄は不変集合だが、その不変多様体が任意の開領域を稠密に埋め尽くすようなことは起きない。一方この馬蹄を含む２次元空間とは独立に１次元考え、その方向に弱い作用をもつような３次元写像を考えれば３次元馬蹄が得られる。さらにこの付け加えた次元方向にある歪みを与えるとブレンダーの特性である「不変多様体がある開領域で稠密に埋め尽くす」現象が起こる。この性質自体、摂動に耐えうるロバストなものであるから一般的なものである。このブレンダー馬蹄は２次元馬蹄と同様で、それだけではヒストリック挙動やディラック物理測度をもっていない。そこで、このブレンダー馬蹄を保ったまま、そのサドル周期点に関するホモクリニック接触を起こすような写像を考える。ホモクリニック接触をもてば非双曲的になり、その非双曲性を活かしすところが本研究の独自性である。具体的には可算無限回の微小のCr摂動を行い、Birkoff平均が存在しない遊走領域を構成したり、任意の周期のサドル周期点にディラック測度があるような写像をヒストリック挙動を起こす写像の構成を行う。
  Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tokai University, 21K03332
• Geometry and statistics of homoclinic tangency in random dynamical systems
  Grants-in-Aid for Scientific Research
  01 Apr. 2019 - 31 Mar. 2023
  Nakano Yushi
  Various properties of dynamical systems with homoclinic tangency, which play a central role in the study of nonhyperbolic dynamical systems, were studied in their counterparts in random dynamical systems, and progress was made that was not expected at the beginning of the study. In particular, it was found that under absolute continuous noise, statistics that differ significantly from those in the noiseless case appear, such as the existence and finiteness of physical measures and the establishment of related limit theorems. We were also able to discover new phenomena related to homoclinic tangency, such as emergent phenomena and the non-existence of Lyapunov exponents.
  Japan Society for the Promotion of Science, Grant-in-Aid for Early-Career Scientists, Tokai University, 19K14575
• Theoretical study of random dynamical systems through the approach of stochastic processes
  Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Research (Exploratory)
  28 Jun. 2019 - 31 Mar. 2022
  Yano Kouji
  Among the various aspects of random dynamical systems, which are given by adding certain randomness to deterministic dynamical systems, we have paid particular attention to the similarities with various properties of Markov processes in the theory of stochastic processes, and have aimed to mathematically elucidate several particular properties of random dynamical systems that are produced by the complex interplay of the properties of the deterministic dynamical systems. In this research, we have obtained remarkable results on the arcsine and the Darling Kac laws for some random dynamical systems constructed by random choice of interval maps, the aging effects for skew Bessel processes, and the problem of resolution of the sigma fields for action evolutions into the driving noise, the infinite past noise, and the third noise.
  Japan Society for the Promotion of Science, Grant-in-Aid for Challenging Research (Exploratory), Kyoto University, 19K21834
• Emergence of differentiable dynamical systems with historic behavior
  Grants-in-Aid for Scientific Research
  01 Apr. 2018 - 31 Mar. 2022
  Soma Teruhiko
  The aim of this research project is to study emergence of diffeomorphisms. In particular, for diffeomorphisms on two-dimensional manifolds and some subsets of positive Lebesgue measure, our goal was to find a family of forward orbit based at points of the subset the emergency of which is Sup-P. The original plan has almost achieved and the result was published in an international journal. Through this research, it was found that the binary code corresponding to the orbit is more essential than diffeomorphism itself. So, we could prove that the above result holds for diffeomorphisms on manifolds of dimension 3 or more.
  Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 首都大学東京, 18K03376
• Historic behavior of wandering domains for high dimensional dynamics
  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
  01 Apr. 2017 - 31 Mar. 2021
  Kiriki Shin
  Results were obtained that largely satisfied the original research objectives. Specifically, we have succeeded to show that C1-generically for diffeomorphisms of manifolds of dimension d at least 3, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behaviour. The results were published in one of the leading journals of the American Mathematical Society.
  Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Tokai University, 17K05283
• 転移作用素のスペクトル構造の解析による確率安定性の研究
  科学研究費助成事業 特別研究員奨励費
  2011 - 2013
  中野 雄史
  今年度は、昨年度に引き続き、(1)準古典解析的手法による部分拡大写像の転移作用素のスペクトル構造の安定性解析、および(2)混合的でも可逆でもない摂動に対する拡大写像の転移作用素のスペクトル安定性を通した確率安定性の研究を行い、その結果を研究集会発表・論文投稿した。
  (1)について、報告者は、昨年度までの研究における最も大きな問題であった「転移作用素(から誘導される擬微分作用素)の主表象計算による評価における、(準古典パラメータに関する)剰余項のノイズ・パラメータに対する非一様性」を、上記の性質がノイズ・パラメータに関して一様に成り立つことを証明する形で解決した。ここでは二重表象計算と呼ばれる技術(森本芳則教授・京都大から示唆していただいた)を主な道具として利用しされたが、これは従来の漸近展開公式では得られなかったノイズ・パラメータに関する一様評価を得るための鍵となった。この結果は、「部分双曲力学系の典型例である2次元トーラス上の部分拡大写像の、SRB測度の唯一性の仮定の下での、確率安定性や相関関数の減衰速度の安定性」などの重要な結論を意味し、研究実施計画にあるようなスペクトル解析による確率安定性の証明においても重要な役割を果たすことを確認した。以上の結果はJ. Wittstenとの共同研究である。
  (2)について、報告者は昨年度までの研究の中で導入された、ランダムな転移作用素(という連続作用素値の確率変数)から誘導されるグラフ変換のスペクトル解析を発展させることで、拡大写像の確率安定性の証明におけるノイズへの混合性・可逆性の仮定を除去することに成功した。これらの仮定は物理的な文脈から考えると不自然な仮定であるものの、技術的な理由から改良が難しかった。そこで、先述の作用素と適切な関数空間を導入することにより、混合的でも可逆でもない摂動下での拡大写像の確率安定性の簡潔な証明を与えることに成功した。
  日本学術振興会, 特別研究員奨励費, 京都大学, 11J01842