Researcher Database
Last Updated :2024/11/14
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Profile and Settings
Affiliation
Degree
Profile and Settings
Name (Japanese)
NAKANOName (Kana)
YushiName
201501001162832699
Affiliation
Achievement
Research Interests
- 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
Research Areas
Research Experience
- 2023/04 - 2024/03 Tokai University School of Science Department of Mathematics Associate Professor
- 2018/04 - 2023/03 Tokai University School of Science Department of Mathematics Junior Associate Professor
- 2016/03 - 2018/03 Kitami Institute of Technology Faculty of Engineering Assistant Professor
- 2015/04 - 2016/03 Osaka City University Osaka Central Advanced Mathematical Institute Researcher
Education
- 2011/04 - 2015/03 Kyoto University Graduate School of Human and Environmental Studies Doctoral course
- 2009/04 - 2011/03 Kyoto University Graduate School of Human and Environmental Studies Master’s course
- 2005/04 - 2009/03 Kyoto University Faculty of Integrated Human Studies
Published Papers
- Masayuki Asaoka, Yushi Nakano, Paulo Varandas, Tomoo Yokoyama2024/11/04In 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 $M$.
- Shin Kiriki, Xiaolong Li, Yushi Nakano, Teruhiko Soma, Edson Vargas2024/04/27We 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 $\mathcal{U}_0$.
- Shin Kiriki, Yushi Nakano, Teruhiko Soma2024/03/30This 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 $n\geq 3$.
- Harry Crimmins, Yushi NakanoErgodic Theory and Dynamical Systems 0143-3857 2023/06/20 [Refereed]
For smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gou\"ezel, Keller, and Liverani, which has been applied to deterministic systems with great success, may be adapted to study random systems that possess good mixing properties. As a consequence, we obtain general linear and higher-order response results for random dynamical systems that we then apply to random Anosov diffeomorphisms and random U(1) extensions of expanding maps. We emphasise that our results apply to random dynamical systems over a general ergodic base map, and are obtained without resorting to infinite dimensional multiplicative ergodic theory. - Shin Kiriki, Yushi Nakano, Teruhiko SomaNonlinearity 36 (8) 4007 - 4033 0951-7715 2023/06/16 [Refereed]
We give a contribution to Takens' last problem [Tak08] in the $C^{1}$ topology (note that solutions have been already given in the smooth category of more than or equal to $C^{2}$, see [KS17,BB]). To be specific, we present a one-parameter family $\mathscr{A}$ of $C^1$ diffeomorphisms having wild blender-horseshoes such that every $C^1$ neighbourhood of any element of $\mathscr{A}$ contains two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure associated with a contracting wandering domain. It is a non-trivial extension of Colli-Vargas' model [CV01] to the higher dimensional dynamics with the use of wild blender-horseshoes. - Yushi Nakano, Teruhiko Soma, Kodai YamamotoDiscrete and Continuous Dynamical Systems 1078-0947 2023/04/01 [Refereed]
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). - Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa, Kouji YanoNonlinearity 36 (3) 1491 - 1509 0951-7715 2023/03/01 [Refereed]
In their recent paper [8], G.Hata and the fourth author first gave an example of random iterations of two piecewise linear interval maps without (deterministic) indifferent periodic points for which the arcsine law -- a characterization of intermittent dynamics in infinite ergodic theory -- holds. The key in the proof of the result is the existence of a Markov partition preserved by each interval maps. In the present paper, we give a class of random iterations of two interval maps without indifferent periodic points but satisfying the arcsine law, by introducing a concept of core random dynamics. As applications, we show that the generalized arcsine law holds for generalized Hata-Yano maps and piecewise linear versions of Gharaei-Homburg maps, both of which do not have a Markov partition in general. - Yong Moo Chung, Yushi Nakano, Jens WittstenDISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 43 (1) 338 - 377 1078-0947 2023/01/04 [Refereed]
The Lyapunov spectra of random U(1) extensions of expanding maps on the torus were investigated in our previous work [NW2015]. Using the result, we extend the recent spectral approach for quenched limit theorems for expanding maps [DFGV2018] and hyperbolic maps [DFGV2019] to our partially hyperbolic dynamics. Quenched central limit theorems, large deviations principles and local central limit theorems for random U(1) extensions of expanding maps on the torus are proved via corresponding theorems for abstract random dynamical systems. - 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) 2189-5384 2023
- Yuta Michimoto, Yushi Nakano, Hisayoshi Toyokawa, Keisuke Yoshida2022/12/30We 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.
- Pablo G. Barrientos, Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa2022/09/19For 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.
- Shin Kiriki, Yushi Nakano, Teruhiko SomaAdvances in Mathematics 400 0001-8708 2022/05/14 [Refereed]
Inspired by a recent work by Berger, we introduce the concept of pointwise emergence. This concept provides with a new quantitative perspective into the study of non-existence of averages for dynamical systems. We show that high pointwise emergence on a large set appears for abundant dynamical systems: Any continuous maps on a compact metric space with the specification property have super-polynomial pointwise emergence on a residual subset of the state space. Furthermore, there is a dense subset of any Newhouse open set each element of which has super-polynomial pointwise emergence on a positive Lebesgue measure subset of the state space. - Shin Kiriki, Xiaolong Li, Yushi Nakano, Teruhiko SomaCommunications in Mathematical Physics 391 (3) 1241 - 1269 0010-3616 2022/05 [Refereed]
Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke [OY2008], whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas [CV2001], as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. We can construct such positive Lebesgue measure sets both as the time averages exist and do not exist on it. - 中野 雄史数理解析研究所講究録 (2217) 144 - 152 1880-2818 2022/04
- Nakano, YushiRIMS Kokyuroku 京都大学数理解析研究所 2176 50 - 56 1880-2818 2021/04
- Nakano YushiRIMS Kokyuroku 京都大学数理解析研究所 2181 (2181) 103 - 111 1880-2818 2021/04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．
- Fumihiko Nakamura, Yushi Nakano, Hisayoshi ToyokawaDISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B 27 (12) 7657 - 7669 1531-3492 2021/03/22 [Refereed]
It has been recently realized that for abundant dynamical systems on a compact manifold, the set of points for which Lyapunov exponents fail to exist, called the Lyapunov irregular set, has positive Lebesgue measure. In the present paper, we show that under any physical noise, the Lyapunov irregular set has zero Lebesgue measure and the number of such Lyapunov exponents is finite. This result is a Lyapunov exponent version of Ara\'{u}jo's theorem on the existence and finitude of time averages. Furthermore, we numerically compute the Lyapunov exponents for a surface flow with an attracting heteroclinic connection, which enjoys the Lyapunov irregular set of positive Lebesgue measure, under a physical noise. This paper also contains the proof of the unique existence of the Lyapunov exponents for a surface flow with an attracting homoclinic/heteroclinic connection under a non-physical noise. - Pablo G. Barrientos, Yushi Nakano, Artem Raibekas, Mario RoldanDYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL 37 (2) 181 - 210 1468-9367 2021/03/20 [Refereed]
We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research. - Fumihiko Nakamura, Yushi Nakano, Hisayoshi ToyokawaNONLINEARITY 35 (1) 66 - 83 0951-7715 2020/09/28 [Refereed][Not invited]
We consider generalized definitions of mixing and exactness for random dynamical systems in terms of Markov operator cocycles. We first give six fundamental definitions of mixing for Markov operator cocycles in view of observations of the randomness in environments, and show that they can be reduced into two different groups. Secondly, we give the definition of exactness for Markov operator cocycles and show that Lin's criterion for exactness can be naturally extended to the case of Markov operator cocycles. Finally, in the class of asymptotically periodic Markov operator cocycles, we show the Lasota-Mackey type equivalence between mixing, exactness and asymptotic stability. - Yushi Nakano, Agnieszka ZelerowiczNONLINEARITY 34 (11) 7609 - 7632 0951-7715 2020/09/01 [Refereed]
In their recent paper [KNS2019], the first author, S. Kiriki, and T. Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In this paper we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any H\"older continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property. This is a natural generalization of [FP2011] to pointwise emergence and Carath\'eodory dimension. - Pablo G. Barrientos, Shin Kiriki, Yushi Nakano, Artem Raibekas, Teruhiko SomaProceedings of the American Mathematical Society 148 (3) 1195 - 1206 0002-9939 2020 [Refereed][Not invited]
© 2020 American Mathematical Society. All rights reserved. We show that C1-generically for diffeomorphisms of manifolds of dimension d ≥ 3, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behavior. - Yushi Nakano, Kenichiro YamamotoTOKYO JOURNAL OF MATHEMATICS 44 (2) 495 - 506 0387-3870 2019/12/27 [Refereed][Not invited]
For any transitive piecewise monotonic map for which the set of periodic
measures is dense in the set of ergodic invariant measures (such as linear mod
$1$ transformations and generalized $\beta$-transformations), we show that the
set of points for which the Birkhoff average of a continuous function does not
exist (called the irregular set) is either empty or has full topological
entropy. This generalizes Thompson's theorem for irregular sets of
$\beta$-transformations, and reduces a complete description of irregular sets
of transitive piecewise monotonic maps to Hofbauer-Raith problem on the density
of periodic measures. - Yushi Nakano, Tomoo YokoyamaCommunications in Mathematical Physics 372 (2) 367 - 383 0010-3616 2019/12/01 [Refereed][Not invited]
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. Since the pioneering work of Ghys et al., it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory to foliations: It has recently been realised that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies. In this paper, we introduce the concept of length averages to singular foliations, and attempt to collect interesting examples with/without length averages. In particular, we demonstrate that length averages exist everywhere for any codimension one C1 orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi-minimal sets of the foliation, which is in strong contrast to time averages of surface flows. - Yushi Nakano, Shota SakamotoDiscrete and Continuous Dynamical Systems- Series A 39 (4) 1779 - 1797 1078-0947 2019/04 [Refereed][Not invited]
© 2019 American Institute of Mathematical Sciences. All Rights Reserved. A typical approach to analysing statistical properties of expanding maps is to show spectral gaps of associated transfer operators in adapted function spaces. The classical function spaces for this purpose are Hölder spaces and Sobolev spaces. Natural generalisations of these spaces are Besov spaces, on which we show a spectral gap of transfer operators. - Shin Kiriki, Yushi Nakano, Teruhiko SomaNonlinearity 32 (3) 1111 - 1124 0951-7715 2019/02/13 [Refereed][Not invited]
© 2019 IOP Publishing Ltd & London Mathematical Society. We consider a parametrised perturbation of a C r diffeomorphism on a closed smooth Riemannian manifold with r ≥ 1, modeled by nonautonomous dynamical systems. A point without time averages for a (nonautonomous) dynamical system is said to have historic behaviour. It is known that for any C r diffeomorphism, the observability of historic behaviour, in the sense of the existence of a positive Lebesgue measure set consisting of points with historic behaviour, disappears under absolutely continuous, independent and identically distributed (i.i.d.) noise. By contrast, we show that the observability of historic behaviour can appear by a non-i.i.d. noise: we consider a contraction mapping for which the set of points with historic behaviour is of zero Lebesgue measure and provide an absolutely continuous, non-i.i.d. noise under which the set of points with historic behaviour is of positive Lebesgue measure. - Shin Kiriki, Yushi Nakano, Teruhiko SomaNonlinearity 30 (8) 3255 - 3270 0951-7715 2017/07/21 [Refereed][Not invited]
© 2017 IOP Publishing Ltd & London Mathematical Society. We present a sufficient condition for three-dimensional diffeomorphisms having heterodimensional cycles to be approximated arbitrarily well by diffeomorphisms with non-trivial contracting wandering domains via several perturbations. The key idea is to show that diffeomorphisms with heterodimensional cycles associated with saddle points with non-real eigenvalues can be approximated by diffeomorphisms with generalized homoclinic tangencies presented by Tatjer. The generalized homoclinic tangency is an organizing center including a Bogdanov-Takens bifurcation, by which one can obtain non-trivial contracting wandering domains together with a Denjoy-like construction. - Yushi NakanoTokyo Journal of Mathematics 40 (1) 165 - 184 0387-3870 2017/06 [Refereed][Not invited]
F. Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle. We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations. The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps. - 中野 雄史数理解析研究所講究録 京都大学数理解析研究所 2028 (2028) 81 - 90 1880-2818 2017/05 [Not refereed][Not invited]
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. - Yushi NakanoStochastics and Dynamics 16 (4) 0219-4937 2016/08/01 [Refereed][Not invited]
© 2016 World Scientific Publishing Company. We consider small perturbations of expanding maps induced by skew-product mappings whose base dynamics need not be mixing or invertible. Adapting a previously developed perturbative spectral approach, we show stability of the densities of the unique absolutely continuous invariant probability measures for expanding maps under these perturbations, and upper bounds on the rate of exponential decay of fiber correlations associated to the measures as the noise level goes to zero. - Yushi Nakano, Masato Tsujii, Jens WittstenNonlinearity 29 (7) 1917 - 1925 0951-7715 2016/05/25 [Refereed][Not invited]
© 2016 IOP Publishing Ltd & London Mathematical Society Printed in the UK. This paper concerns the compact group extension f : ⌉2→⌉2, f (x, s) = (E(x), s + τ(x) mod 1) of an expanding map E : S1→S1. The dynamics of f and its stochastic perturbations have previously been studied under the so-called partial captivity condition. Here we prove a supplementary result that shows that partial captivity is a C τ generic condition on τ, once we fix E. - 中野 雄史数理解析研究所講究録 京都大学数理解析研究所 1942 (1942) 66 - 77 1880-2818 2015/04 [Not refereed][Not invited]
- Yushi Nakano, Jens WittstenNonlinearity 28 (4) 951 - 1002 0951-7715 2015/04 [Refereed][Not invited]
© 2015 IOP Publishing Ltd & London Mathematical Society. We consider quenched random perturbations of skew products of rotations on the unit circle over uniformly expanding maps on the unit circle. It is known that if the skew product satisfies a certain condition (shown to be generic in the case of linear expanding maps), then the transfer operator of the skew product has a spectral gap. Using semiclassical analysis we show that the spectral gap is preserved under small random perturbations. This implies exponential decay of quenched random correlation functions for smooth observables at small noise levels.
Research Projects
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2023/04 -2027/03Author : 中野 雄史
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)Date (from‐to) : 2022/04 -2026/03Author : 相馬 輝彦, 桐木 紳, 中野 雄史
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)Date (from‐to) : 2021/04 -2026/03Author : 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。
- 日本学術振興会：科学研究費助成事業 若手研究Date (from‐to) : 2019/04 -2023/03Author : 中野 雄史本研究課題は、ランダムなホモクリニック接触を持つ力学系を深く理解することを最終的な目標としているが、これに関連して事業初年度では、(1)ランダム力学系の周期的挙動、および(2)ホモクリニック接触を持つような力学系の研究を行った。 (1)については、中村文彦氏（北見工大）と共同で（ランダム力学系の転送作用素を含む）Markov作用素コサイクルの漸近周期性に関する2種類の定義を導入し、それらの差異が現れる例の構成とそれらの周期性が得られるためのスペクトル論的および幾何的な十分条件を得た。特に（決定論的）周期が異なる2つの写像をランダムに反復合成した力学系においては、相空間が2つ以上の既約空間に分解されるものの、非周期的な挙動を見せる例が豊富にあることがわかり、それゆえ前述の十分条件にはノイズ側である種の位相的な仮定が必要であることが判明した。この結果は投稿に向けて準備中である。他にもランダム力学系の転送作用素に関して、急冷型極限定理に関する結果を鄭容武氏・J. Wittsten氏との、準安定性に関する結果をS. Lloyd氏との、線形反応問題に関する結果をH. Crimmins氏との共同研究にて得た。 (2)については、ホモクリニック接触を持つ力学系を微小摂動すると、時間平均が存在しないような点の集合（非正則集合）がLebesgue測度正となる例を豊富に作れることが知られているが、相馬輝彦氏（東京都立大）および桐木紳氏（東海大）と共同でこの手法を拡張し、創発現象に関するBerger予想の部分的解決に成功した。この結果は現在投稿中である。また、関連してP. Berger氏、S. Bielber氏と研究交流を持つこととなり、日本およびフランスで情報交換を行った。さらに、A. Zelerowicz氏と先述の結果を熱力学形式の文脈に拡張することに成功し、現在これを投稿に向けて準備中である。
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Research (Exploratory)Date (from‐to) : 2019/06 -2022/03Author : 矢野 孝次, 佐藤 譲, 角 大輝, 中野 雄史一般化逆正弦法則の発展的問題として，秋元（東京理科大学）および大学院生の世良と山戸と共同で，エイジングの問題に取り組み始めた．原点出発のブラウン運動の滞在時間分布は逆正弦法則と呼ばれるが，その一般化として歪ベッセル過程の滞在時間分布はランペルティ分布に従うことがよく知られている．これらの結果は出発時点から観測を開始したときのものであるが，観測開始時間を零から正に変えたときどのような影響が現れるかを調べるのがエイジングの問題である．ブラウン運動および歪ベッセル過程に対して詳しく調べ，理論物理的視点でまとめた結果を投稿するに至った． 作用発展に対する情報系分解問題について，伊藤（京都産業大学）および大学院生の世良と共同で，有限集合上のランダム写像反復モデルに対して一般的な結果をまとめることができた．有限集合を状態空間とし，ランダム写像反復による作用発展を考えるとき，一粒子のノイズには駆動ノイズと無限過去ノイズの他に第三ノイズが生ずることが分かっていたが，これらのノイズを成分としてどのように構成されているかは不明であった．本研究では多粒子のノイズを観測することによりこの問題を解決した．リース分解を用いた半群上のランダム変数の無限積に関する既知の一般的結果に基づき，情報系の結合演算の厳密な取り扱いに注意を払いながら，作用発展の情報系を，駆動ノイズ，無限過去ノイズ，第三ノイズに分解する公式を得ることができた．この結果は投稿に向けて準備中である．
- 日本学術振興会：科学研究費助成事業 基盤研究(C)Date (from‐to) : 2018/04 -2022/03Author : 相馬 輝彦, 桐木 紳, 中野 雄史本研究課題は，微分同相写像の創発性(emergence)である．特に，2次元多様体 M 上の微分同相写像で，Lebesgue 測度が正である M のある部分集合 U に対し，U の要素を起点とする前方軌道の創発性が Sup-P となる族を見つけることにあった．この研究は，創発性の概念を導入した P. Berge氏 (2017) の結果が本研究代表者達の発表した論文(2017)と密接に関連していることが動機となっている．本研究課題の研究代表者（相馬）は，研究者分担者の桐木紳氏（東海大学教授），中野雄史氏（東海大学講師）と共同でこの研究に取り組み，本研究課題２年目の目標はほぼ達成できた．これらの結果を３名の共著論分「Emergence via via non-existence of averages」としてまとめ，現在投稿中である．前年度得られた結果は「Sup-P 創発性」に関するものであったが，今年度の結果はさらに強い「stretched 創発性」に関するものである．具体的には次の結果が得られた．
【定理】微分同相空間 Diff(M) 内の任意の Newhouse 開集合の稠密な部分集合 D で次の性質を持つものが存在する．「f を D の要素とするとき，M 内に Lebesgue 測度正の部分集合が存在し，その部分集合の要素を起点とする前方軌道は pointwise stretched 創発性をもつ．」
【定理】f を m 個の文字の両側無限列空間 X 上の左フル・シフトとする．このとき，X の residual な部分集合 R で次の性質を持つものが存在する．R の要素を起点とする前方軌道は pointwise stretched 創発性をもつ． - Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)Date (from‐to) : 2017/04 -2021/03Author : Kiriki ShinResults 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.
- 日本学術振興会：科学研究費助成事業 特別研究員奨励費Date (from‐to) : 2011 -2013Author : 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。