Tsujii Masato | Last modified date：2021.06.13 |

Professor /
Department of Mathematics /
Faculty of Mathematics

**Presentations**

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, $|Re(s)|< au$, and the region $Re(s)<-chi_0+ au$, 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.. |

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.. |

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