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Tsujii Masato Last modified date:2023.12.06

Graduate School
Undergraduate School
Administration Post
Dean of the Faculty of Mathematics
Dean of the Graduate School of Mathematics

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Academic Degree
Doctoral degree of Science
Country of degree conferring institution (Overseas)
Field of Specialization
ORCID(Open Researcher and Contributor ID)
Total Priod of education and research career in the foreign country
Outline Activities
My research area is dynamical system theory(the qualitative theory of ordinary differential equations) and ergodic theory. I study transfer operators associated to chaotic dynamical systems by using methods of functional analysis such as Fourier analysis and semi-classical analysis. My main target of research now is the geodesic flows on negatively curved manifold, which is a type of chaotic hyperbolic flow, and the corresponding transfer operators and semi-calsscial zeta funtions.

From April 2022, I have been working as the dean of Faculty of Mathematics, Graduate school of Mathematics and Joint Graduate school of Mathematics-for-Innovation.

For education, I gave a few lectures such as calculus, linear algebra, comlex function theory, dynamical system theory, etc) and supervised a few students in their studies of mathematics.
Research Interests
  • Exponential mixing of Anosov flows and problems in quantum chaos
    keyword : Quantum chaos, Anosov flow, geodesic flow
  • Spectral study of transfer operators for hyperbolic dynamical systems
    keyword : Dynamical Systems, Transfer operators
  • Transfer operators for geodesic flows and the semi-classical zeta function
    keyword : geodesic flow, transfer operators, the semi-classical zeta functions
  • Functional analytic methods in dynamical systems theory
    keyword : hypebolic dynamical systems, transfer operators, dynamical zeta functions
Academic Activities
1. Masato Tsujii, Zhiyuan Zhang, Smooth mixing Anosov flows in dimension three are exponentially mixing., Annals of Math., 197, 1, 65-158, 2023.01, 3次元のアノソフ流が位相混合的であるならば指数混合的であることを示した。.
2. Carlene Kalle, Valentin Matache, Evgeny Verbitskiy, Invariant densities for random continued fractions, Journal of Mathematical Analysis and Applications, Vol. 512, No.2, (2022), 512, 2, 2022.05.
3. Masato Tsujii, Geodesic flows of negatively curved manifolds and the semi-classical zeta function, Sugaku, 10.1090/suga/429, 18, 69-92, 2018.10.
4. 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, 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..
5. 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, アノソフ流は生成的な条件の下で指数混合的であろうという長年の予想について、3次元の体積保存系の場合に肯定的に解決した。.
6. Tsujii Masato, On the Fourier transforms of self-similar measures, Dynamical Systems – an international journal, 2015.11.
7. 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..
8. Tsujii Masato, Geodesic flows on negatively curved manifolds and the semi-classical zeta function, 日本数学会, 2014.07.
9. 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.
10. Tsujii Masato, Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform, Ergodic theory and Dynamical systems (Cambridge University Press),, 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..
11. 辻井 正人, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 2010.02.
12. 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.
13. 辻井 正人, Decay of correlations in suspension semi-flow of angle-multiplying maps, Ergodic Therory and Dynamical Systems (Cambridge), 2007.12.
1. 辻井 正人(本人), 3-dimensional Anosov flows, Spring School on Transfer operator, 2021.03, [URL], In this mini-course, I am going to discuss mainly about local geometric structure of the(strong) stable and unstable lamination of smooth Anosov flows, particularly in the lowest dimensional case, that is, in dimension 3. We will also explain how that structure leads to (exponential) mixing property of the flow, using a simplified model of the U(1)-extensions of Anosov diffeomorphisms. The content will be based on my joint work with Zhiyuan Zhang..
2. Masato Tsujii, COHOMOLOGICAL THEORY OF THE SEMI-CLASSICAL ZETA FUNCTIONS, Dynamics, Equations and Applications 2019, 2019.09, We first review very briefly about recent developments in analysis of transfer operators for hyperbolic dynamical systems. We will then focus on the semi-classical (or Gutzwiller-Voros) zeta functions for geodesic flows on negatively curved manifolds. We show that the semi-classical zeta function is the dynamical Fredholm determinant of a transfer operator acting on the leaf-wise cohomology space along the unstable foliation. This realize the idea presented by Guillemin and Patterson a few decades ago. As an application, we see that the zeros of the semi-classical zeta function concentrate along the imaginary axis, imitating those of Selberg zeta function..
3. 辻井 正人, Large Deviations Principle for S-unimodal maps, Real and Complex dynamics of Henon maps, 2019.11, [URL], I will speak about a recent joint-work with Hiroki Takahasi (Keio Univ.), in which we proved the Large Deviations Principle of Level 2 (LDP2) for ALL S-unimodal maps. This work is based on a recent work by Chung, Rivera-Letelier and Takahasi, where the LDP2 is proved for any non- renormalizable S-unimodal maps. To extend LDP2 to all S-unimodal maps, we provided some analysis on the process where orbits on a renormalization cycle fall in a deeper cycle. I will also give a counter-example of a bi-modal cubic map for which LDP2 does not hold..
4. Masato Tsujii, Vivian Baladi, Transfer operators for Anosov diffeomorphisms, Beyond Uniform Hyperbolicity 2019, 2019.07, We present a functional-analytic approach to the study of transfer operators for Anosov flows. To study transfer operators, a basic idea in semi-classical analysis suggests to look at the action of the flow on the cotangent bundle. Though this idea is simple and intuitive (as we will explain in the lectures), we need some framework to make it work. In the lectures, we present such a framework based on a wave-packet transform..
5. 辻井 正人, Exponential mixing for volume-preserving Anosov flows, Real and Complex dynamics of Henon maps, 2019.03.
6. 辻井 正人, On cohomological theory of dynamical zeta functions, Dynamical Systems and Related Topics, 2018.08.
7. 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.
8. 辻井 正人, On cohomological theory of dynamical zeta functions, Spectral geometry, graphs and semiclassical analysis, 2017.12.
9. 辻井 正人, Exponential decay of correlations for Anosov flows, Analytical aspects of hyperbolic flows, 2017.07.
10. 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.
11. 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.
12. 辻井 正人, Exponential mixing for generic volume-preserving Anosov flows in dimension three, Mixing flows and averaging method, 2016.04, [URL].
13. 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, [URL].
14. Tsujii Masato, The error term of The prime Orbit Theorem for expanding semi-flows, School and Conference on Dynamical Systems, 2015.08, [URL].
15. 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.
16. 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..
17. 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 $ au>0$, that its zeros are contained in the union of the $ au$-neighborhood of the imaginary axis, $|Re(s)|0$ 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..
18. 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..
19. 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..
Membership in Academic Society
  • Japan Mathematical Society
Educational Activities
As I have been the dean of Faculty of Mathematics, I am not in charges of lectures. I am supervising 3 master and 2 doctoral course students.
Professional and Outreach Activities
I attended as a lecturer to TMU-ICTP School and Conference on Dynamical Systems and Ergodic Theory at Tarbiat Modares University. .