Tsujii Masato | Last modified date：2020.06.22 |

Graduate School

Undergraduate School

Homepage

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personal webpage .

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English version .

Academic Degree

Doctoral degree of Science

Country of degree conferring institution (Overseas)

No

Field of Specialization

Mathematics

ORCID(Open Researcher and Contributor ID)

https://orcid.org/0000-0002-2332-0870

Total Priod of education and research career in the foreign country

03years00months

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.

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.

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

**Research Interests**

- Spectral study of transfer operators for hyperbolic dynamical systems

keyword : Dynamical Systems, Transfer operators

2016.04～2020.03. - Transfer operators for geodesic flows and the semi-classical zeta function

keyword : geodesic flow, transfer operators, the semi-classical zeta functions

2010.04～2015.03. - Functional analytic methods in dynamical systems theory

keyword : hypebolic dynamical systems, transfer operators, dynamical zeta functions

2006.04～2010.03.

**Academic Activities**

**Papers**

**Presentations**

1. | 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.. |

2. | 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.. |

3. | 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.. |

4. | 辻井 正人, Exponential mixing for volume-preserving Anosov flows, Real and Complex dynamics of Henon maps, 2019.03. |

5. | 辻井 正人, On cohomological theory of dynamical zeta functions, Dynamical Systems and Related Topics, 2018.08. |

6. | 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. |

7. | 辻井 正人, On cohomological theory of dynamical zeta functions, Spectral geometry, graphs and semiclassical analysis, 2017.12. |

8. | 辻井 正人, Exponential decay of correlations for Anosov flows, Analytical aspects of hyperbolic flows, 2017.07. |

9. | 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. |

10. | 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. |

11. | 辻井 正人, Exponential mixing for generic volume-preserving Anosov flows in dimension three, Mixing flows and averaging method, 2016.04, [URL]. |

12. | 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]. |

13. | Tsujii Masato, The error term of The prime Orbit Theorem for expanding semi-flows, School and Conference on Dynamical Systems, 2015.08, [URL]. |

14. | 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. |

15. | 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.. |

16. | 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. . |

17. | 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.. |

18. | 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. . |

19. | Decay of correlations and dynamical zeta functions for hyperbolic flows. |

**Membership in Academic Society**

- Japan Mathematical Society

**Awards**

- Exponential mixing for volume-preserving Anosov flows in dimension 3.
- Functional analytic methods in smooth ergodic theory.

Educational

**Educational Activities**

I gave two courses in mathematics department (complex analysis and dynamical system theory) and seminar course for 2 undergraduates and 1 graduate.

Social

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