九州大学 研究者情報
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基本情報 研究活動 教育活動 社会活動
辻井 正人(つじい まさと) データ更新日:2019.04.25



主な研究テーマ
測地流の転移作用素と半古典ゼータ関数の研究
キーワード:測地流,転移作用素,半古典ゼータ関数
2010.04~2015.03.
力学系理論における関数解析的方法
キーワード:双曲力学系,転移作用素,力学系のゼータ関数
2006.04~2010.03.
研究業績
主要著書
1. Hirsh, Smale, Devaney 辻井,桐木,谷川,三波, 力学系入門(原著第2版), 共立出版, 2007.08.
主要原著論文
1. Masato Tsujii, The error term in the prime orbit theorem for expanding semiflows, Ergodic Theory and Dynamical Systems, 10.1017/etds.2016.113, 38, 5, 1954-2000, 2018.08, [URL], We consider suspension semiflows of angle-multiplying maps on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number of prime periodic orbits with period . The error term is bounded, at least, by for arbitrarily small ϵ>0, where and are, respectively, the topological entropy and the maximal Lyapunov exponent of the semiflow..
2. Tsujii Masato, Exponential mixing for generic volume-preserving Anosov flows in dimension three, Journal of the Mathematical Society of Japan, 10.2969/jmsj/07027595, 70, 2, 757-821, 2018.04.
3. Tsujii Masato, Faure Frederic, The semiclassical zeta function for geodesic flows on negatively curved manifolds, Inventiones Mathematicae, 10.1007/s00222-016-0701-5, 208, 3, 851-998, 2017.06, We consider the semi-classical (or Gutzwiller-Voros) zeta function for C∞ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any τ>0, that its zeros are contained in the union of the τ-neighborhood of the imaginary axis, |ℜ(s)|<τ, and the region ℜ(s)<−χ+τ, up to finitely many exceptions, where χ0>0 is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law..
4. Tsujii Masato, On the Fourier transforms of self-similar measures, Dynamical Systems – an international journal, 2015.11.
5. Frédéric Faure, Tsujii Masato, Prequantum transfer operator for symplectic anosov diffeomorphism, Asterisque, 2015-January, 375, 1-237, 2015.01, We define the preauantization of a symplectic Aaosov diffeoniorphism f : M -∗ M as a U(l) extension of the diffeoniorphism / preserving a connection related to the symplectic structure on M. We study the spectral properties of the associated transfer operator with a given potential V € C°° (M), called prequantum transfer operator. This is a model of transfer operators for geodesic flows on negatively curved manifolds {or contact Anosov flows). We restrict the prequantum transfer operator to the JV-the Fourier mode with respect to the U(l) action and investigate the spectral property in the limit N -∗ oo, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, under some pinching conditions, we show a "band structure" of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin. We show that, with the special (Holder continuous) potential Vo = 1/2log|det Df|Eu|1, where £ is the unstable subspace, the outermost annulus is the unit circle and separated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponentially small reminder for large time. We show also that, for a potential V such that the outermost annulus is separated from the other parts, most of the eigenvalues in the outermost annulus concentrate on a circle of radius exp((V - V0)) where (.) denotes the spatial average on M. The number of the eigenvalues in the outermost annulus satisfies a Weyl law, that is, NdVol (M) in the leading order with d = 1/2dimM. We develop a semiclassical calculus associated to the prequantum operator by defining quantization of observables OpN (Ψ) as the spectral projection of multiplication operator by Ψ to this outer annulus. We obtain that the semiclassical Egorov formula of quantum transport is exact. The correlation functions defined by the classical transfer operator are governed for large time by the restriction to the outer annulus that we call the quantum operator. We interpret these results from a physical point of view as the emergence of quantum dynamics in the classical correlation functions for large..
6. Tsujii Masato, Geodesic flows on negatively curved manifolds and the semi-classical zeta function, 日本数学会, 2014.07.
7. Tsujii Masato, Band structure of the Ruelle spectrum of contact Anosov flows, COMPTES RENDUS MATHEMATIQUE, 10.1016/j.crma.2013.04.022, 351, 9-10, 385-391, 2013.05.
8. Tsujii Masato, Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform, Ergodic theory and Dynamical systems (Cambridge University Press), http://dx.doi.org/10.1017/S0143385711000605, 32, 6, 2083-2118, 2012.10, This paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radii of the transfer operators acting on an appropriate function space exactly and improves the previous result in Tsujii [Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23 (2010), 1495–1545]. Also, we provide a simplified proof by using the so-called Fourier–Bros–Iagolnitzer (FBI) (or Bargmann) transform..
9. 辻井 正人, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 2010.02.
10. Viviane Baladi, Masato TSUJII, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, "Probabilistic and Geometric Structures in Dynamics", K. Burns, D. Dolgopyat and Ya. Pesin (eds), Contemp. Math. (Amer. Math. Soc.), Volume in honour of M. Brin's 60th birthday, 29-68ページ, 2008.10.
11. 辻井 正人, Decay of correlations in suspension semi-flow of angle-multiplying maps , Ergodic Therory and Dynamical Systems (Cambridge), 2007.12.
主要学会発表等
1. 辻井 正人, Exponential mixing for volume-preserving Anosov flows, Real and Complex dynamics of Henon maps, 2019.03, [URL].
2. 辻井 正人, On cohomological theory of dynamical zeta functions, Dynamical Systems and Related Topics, 2018.08, [URL].
3. 辻井 正人, Gutzwiller-Voros zeta functions for geodesic flows on negatively curved manifolds, The third international conference on the dynamics of differential equations, 2017.12, [URL].
4. 辻井 正人, On cohomological theory of dynamical zeta functions, Spectral geometry, graphs and semiclassical analysis, 2017.12, [URL].
5. 辻井 正人, Exponential decay of correlations for Anosov flows, Analytical aspects of hyperbolic flows, 2017.07, [URL].
6. 辻井 正人, The spectrum of semi-classical transfer operator for expanding-semi flows, Tokyo–Berkeley Mathematics Workshop Partial Differential Equations and Mathematical Physics, 2017.01, [URL].
7. 辻井 正人, Exponential mixing for generic volume-preserving Anosov flows in dimension three, Analytical Methods in Classical and Quantum Dynamical Systems, 2016.07, [URL].
8. 辻井 正人, Exponential mixing for generic volume-preserving Anosov flows in dimension three, Mixing flows and averaging method, 2016.04.
9. 辻井 正人, The spectrum of semi-classical transfer operator for expanding semi-flows with holes, Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism, 2016.03.
10. Tsujii Masato, The error term of The prime Orbit Theorem for expanding semi-flows, School and Conference on Dynamical Systems, 2015.08.
11. Tsujii Masato, Spectrum of transfer operators for expanding semi-flows., 国際数学者会議(ICM2014)サテライト会議 Dynamical systems and related topics, 2014.08.
12. Tsujii Masato, Resonances for geodesic flows on negatively curved manifolds, 国際数学者会議(ICM2014), 2014.08, We report some recent progress in the study of geodesic flows on negatively curved manifolds (or more generally contact Anosov flows). We consider one-parameter groups of transfer operators associated to the flows and investigate the spectra of their generators. The main ingredients are the recent results about a band structure of the discete spectrum, which are obtained in the authors’ joint works..
13. 辻井 正人, Geodesic flows on negatively curved manifolds and the semi-classical zeta function, Measurable and Topological Dynamical Systems,  Keio 2013, 2013.12, [URL], We consider the semi-classical (or Gutzwiller-Voros) zeta functions for $C^\infty$ contact Anosov flows. Analyzing the spectra of the generators for some transfer operators associated to the flow, we prove, for any $\tau>0$, that its zeros are contained in the union of the $\tau$-neighborhood of the imaginary axis, $|\Re(s)|<\tau$, and the region $\Re(s)<-\chi_0+\tau$, up to finitely many exceptions, where $\chi_0>0$ is the hyperbolicity exponent of the flow.
Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law. .
14. Tsujii Masato, Spectrum of geodesic flow on negatively curved manifold, Hyperbolicity and Dimension, 2013.12, [URL], We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides. Those eigenvalues appear as the zeros of the so-called semi-classical (or Gutzwiller-Voros) zeta functions. These results are obtained as application of some ideas in the semi-classical analysis..
15. Tsujii Masato, The semi-classical zeta function, ICTP-ESF School and Conference in Dynamical Systems, 2012.06, We discuss about analytic properties of the so-called semi-classical zeta functions for geodesic flows on the closed manifold with negative sectional curvature. The main results is that the concentration of its zeros along the imaginary axis, which is a generalization of the classical results of Selberg. .
16. Masato TSUJII, Functional analytic method in smooth ergodic theory, Dynamics and PDE, 2010.01.
17. 辻井 正人, 力学系のゼータ関数と転移作用素のスペクトル, Encounter with Mathematics, 2007.05.
18. 辻井 正人, 双曲的な流れにおける相関の減衰と力学系のゼータ関数, 力学系と微分方程式, 2007.11.
19. Masato TSUJII, Quasi-compactness of transfer operators for contact Anosov flows, Workshop GREFI-MEFI "From dynamical systems to statistical mechanics, 2008.02, [URL].
20. Masato TSUJII, Ruelle resonances in contact Anosov flows, NCTS 2008 Workshop on Dynamical systems, 2008.05, [URL].
21. Masato TSUJII, Quasi-compactness of transfer operators for contact Anosov flow, WORKSHOP on HYPERBOLIC DYNAMICAL SYSTEMS WITH SINGULARITIES, 2009.06.
22. Masato TSUJII, On the semi-classical zeta functions for negatively curved manifolds, Mini-Workshop: Spectrum of Transfer Operators: Recent Developments and Applications, 2009.11.
23. Masato TSUJII, Contact Anosov Flows and the FBI transform, Hyperbolic Dynamics in the Sciences, 2010.06.
24. Masato TSUJII, 接触アノソフ流と FBI 変換, 力学系冬の研究集会, 2011.01.
25. Masato TSUJII, 負曲率多様体上の測地流, 談話会, 2011.01.
26. Masato TSUJII, 負曲率多様体上の測地流と半古典ゼータ関数, Dynamics of Complex Systems 2011, 2011.03.
27. Masato TSUJII, Semiclassical zeta functions for negatively curved manifolds, Kyoto Dynamics Days 10, 2011.03.
その他の優れた研究業績
2016.06, 典型的なカオス的は流れ(連続力学系)であるアノソフ流における指数的混合性が生成的な条件の下で成立するかとうかは1970年代のアノソフやシナイの研究依頼の懸案であったが,その問題に対し3次元の場合に肯定的な解決を与えた. 結果と証明は論文 "Exponential mixing for generic volume-preserving Anosov flows in dimension three" (プレプリント, arXiv:1601.00063)で発表した. .
学会活動
所属学会名
日本数学会
学協会役員等への就任
2008.04~2009.03, 日本数学会, 評議員.
学会大会・会議・シンポジウム等における役割
2008.06.16~2008.06.16, Mini-workshop on ergodic theory, 世話人.
2008.05.19~2008.05.22, Flat surfaces and Teichmuller geodesic flows, 世話人.
学会誌・雑誌・著書の編集への参加状況
2013.04~2014.03, Kyushu Journal of mathematics, 国際, 電子化推進委員.
2004.04, Dynamical systems, an international journal, 国際, 編集委員.
2004.04, Ergodic Theory and Dynamical systems, 国際, 編集委員.
2004.04~2018.03, Nonlinearity, 国際, 編集委員.
学術論文等の審査
年度 外国語雑誌査読論文数 日本語雑誌査読論文数 国際会議録査読論文数 国内会議録査読論文数 合計
2018年度
2017年度
2016年度
2016年度
2015年度
2012年度
2011年度
2010年度
2009年度 14  14 
2008年度 14  14 
2007年度 10  11 
その他の研究活動
海外渡航状況, 海外での教育研究歴
Chungnam 大学(韓国 大田), Coex, Korea, 2014.08~2014.08.
Corinaldo, Italy, 2010.05~2010.05.
Mittag Leffeler Institute, Sweden, 2010.01~2010.02.
KTH, Sweden, 2009.12~2009.12.
Fourier Institute, Rome university 3, Ecole Normal, MFO, France, Italy, Germany, 2009.09~2009.11.
Rice University, UnitedStatesofAmerica, 2009.03~2009.04.
NCTS(国家理論科学中心),, Taiwan, 2008.05~2008.05.
CIRM, Luminy, France, 2008.02~2008.02.
中国科学院, China, 2007.06~2007.06.
外国人研究者等の受入れ状況
2018.09~2018.12, 1ヶ月以上, Armenian Academy of Science, Armenia, 日本学術振興会.
2008.10~2008.10, 2週間未満, PUC Rio, Spain, 日本学術振興会.
2008.05~2008.05, 2週間未満, Univ. Renne, Russia, 科学技術振興事業団.
2008.06~2008.06, 2週間未満, Rice University, Russia, Rice University.
受賞
日本数学会 JMSJ論文賞(2018年度), 日本数学会, 2019.03.
日本数学会 秋季賞(2013年度), 日本数学会, 2013.09.
研究資金
科学研究費補助金の採択状況(文部科学省、日本学術振興会)
2015年度~2019年度, 基盤研究(B), 代表, 双曲力学系における転送作用素のスペクトルの研究.
2010年度~2014年度, 基盤研究(B), 代表, 測地流の転移作用素と半古典ゼータ関数の研究.
2006年度~2009年度, 基盤研究(B), 代表, 力学系理論における関数解析的方法の研究.

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