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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.. |

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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。. |

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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.. |

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Yushi Nakano, Tsujii Masato, Jens Wittsten, The partial captivity condition for U(1) extensions of expanding maps on the circle, *Nonlinearity*, 10.1088/0951-7715/29/7/1917, 29, 7, 1917-1925, 2016.05, 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 : S^{1}→S^{1}. 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.. |

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Tsujii Masato, On the Fourier transforms of self-similar measures, *Dynamical Systems – an international journal*, 2015.11. |

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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|E_{u}|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 - V_{0})) where (.) denotes the spatial average on M. The number of the eigenvalues in the outermost annulus satisfies a Weyl law, that is, N^{d}Vol (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.. |

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Tsujii Masato, Geodesic flows on negatively curved manifolds and the semi-classical zeta function, *日本数学会*, 2014.07. |

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Frédéric Faure, Tsujii Masato, Semiclassical approach for the Ruelle-Pollicott spectrum of hyperbolic dynamics, *Springer INdAM Series*, 10.1007/978-3-319-04807-9_2, 65-135, 2014.01, Uniformly hyperbolic dynamics (Axiom A) have “sensitivity to initial conditions” and manifest “deterministic chaotic behavior”, e.g. mixing, statistical properties etc. In the 1970, David Ruelle, Rufus Bowen and others have introduced a functional and spectral approach in order to study these dynamics which consists in describing the evolution not of individual trajectories but of functions, and observing the convergence towards equilibrium in the sense of distribution. This approach has progressed and these last years, it has been shown by V. Baladi, C. Liverani, M. Tsujii and others that this evolution operator (“transfer operator”) has a discrete spectrum, called “Ruelle-Pollicott resonances” which describes the effective convergence and fluctuations towards equilibrium.Due to hyperbolicity, the chaotic dynamics sends the information towards small scales (high Fourier modes) and technically it is convenient to use “semiclassical analysis” which permits to treat fast oscillating functions. More precisely it is appropriate to consider the dynamics lifted in the cotangent space T ^{∗} M of the initial manifold M (this is an Hamiltonian flow). We observe that at fixed energy, this lifted dynamics has a relatively compact non-wandering set called the trapped set and that this lifted dynamics on T ^{∗} M scatters on this trapped set. Then the existence and properties of the Ruelle-Pollicott spectrum enters in a more general theory of semiclassical analysis developed in the 1980 by B. Helffer and J. Sjöstrand called “quantum scattering on phase space”.We will present different models of hyperbolic dynamics and their Ruelle-Pollicott spectrum using this semi-classical approach, in particular the geodesic flow on (non necessary constant) negative curvature surface ℳ. In that case the flow is on M=T1∗ℳ, the unit cotangent bundle of ℳ. Using the trace formula of Atiyah-Bott, the spectrum is related to the set of periodic orbits.We will also explain some recent results, that in the case of Contact Anosov flow, the Ruelle-Pollicott spectrum of the generator has a structure in vertical bands. This band spectrum gives an asymptotic expansion for dynamical correlation functions. Physically the interpretation is the emergence of a quantum dynamics from the classical fluctuations. This makes a connection with the field of quantum chaos and suggests many open questions.. |

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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. |

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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.. |

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辻井 正人, Quasi-compactness of transfer operators for contact Anosov flows, *Nonlinearity*, 2010.02. |

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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. |

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辻井 正人, Decay of correlations in suspension semi-flow of angle-multiplying maps , *Ergodic Therory and Dynamical Systems (Cambridge)*, 2007.12. |