1. |
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. |

2. |
Masato Tsujii, Geodesic flows of negatively curved manifolds and the semi-classical zeta function, *Sugaku*, 10.1090/suga/429, 18, 69-92, 2018.10. |

3. |
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.. |

4. |
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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Frederic Faure, Masato Tsujii, 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 functions for contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small , its zeros are contained in the union of the -neighborhood of the imaginary axis, , and the half-plane , up to finitely many exceptions, where is the hyperbolicity exponent of the flow. Further we show that the density of the zeros along 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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Masato Tsujii, Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform, *ERGODIC THEORY AND DYNAMICAL SYSTEMS*, 10.1017/S0143385711000605, 32, 2083-2118, 2012.12, 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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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.. |

14. |
Masato Tsujii, Quasi-compactness of transfer operators for contact Anosov flows, *NONLINEARITY*, 10.1088/0951-7715/23/7/001, 23, 7, 1495-1545, 2010.07, For any C(r) contact Anosov flow with r >= 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all the C(r) functions, such that the one-parameter family of transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. We also give explicit bounds on the essential spectral radii of those extensions in terms of differentiability r and the hyperbolicity exponents of the flow.. |

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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. |

17. |
Masato Tsujii, Decay of correlations in suspension semi-flows of angle-multiplying maps, *Ergodic Theory and Dynamical Systems*, 10.1017/S0143385707000430, 28, 1, 291-317, 2008.02, We consider suspension semi-flows of angle-multiplying maps on the circle for C^{r} ceiling functions with r3. Under a C^{r}generic condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L^{2} space such that the PerronFrobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the PerronFrobenius operator for the time-t-map is quasi-compact for a C^{r} open and dense set of ceiling functions.. |

18. |
辻井 正人, Decay of correlations in suspension semi-flow of angle-multiplying maps, *Ergodic Therory and Dynamical Systems (Cambridge)*, 2007.12. |

19. |
Viviane Baladi, Masato Tsujii, Anisotropic hölder and sobolev spaces for hyperbolic diffeomorphisms, *Annales de l'Institut Fourier*, 10.5802/aif.2253, 57, 1, 127-154, 2007.01, We study spectral properties of transfer operators for diffeomorphisms T : X → X on a Riemannian manifold X. Suppose that Ω is an isolated hyperbolic subset for T, with a compact isolating neighborhood V ⊂ X. We first introduce Banach spaces of distributions supported on V, which are anisotropic versions of the usual space of C^{p} functions C^{p} (V) and of the generalized Sobolev spaces W^{p,t}(V), respectively. We then show that the transfer operators associated to T and a smooth weight g extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents.. |

20. |
Carlangelo Liverani, Masato Tsujii, Zeta functions and dynamical systems, *NONLINEARITY*, 10.1088/0951-7715/19/10/011, 19, 10, 2467-2473, 2006.10, In this brief paper we present a very simple strategy to investigate dynamical determinants for uniformly hyperbolic systems. The construction builds on the recent introduction of suitable functional spaces which allow us to transform simple heuristic arguments into rigorous ones. Although the results so obtained are not exactly optimal, the straightforwardness of the argument makes them noticeable.. |

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Artur Avila, Sébastien Gouëzel, Masato Tsujii, Smoothness of solenoidal attractors, *Discrete and Continuous Dynamical Systems*, 10.3934/dcds.2006.15.21, 15, 1, 21-35, 2006.05, We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle S ^{1}: T-.S^{1} X ℝ-.S^{1} × ℝ, T(x,y) = (ℓx, λy +f /(x)) where ≥ 2, 0 < λ < 1 and f is a C^{r} function on S^{1}. We show that, if λ ^{1+2s} > 1 for some 0 * s < r - 2, the density of the SBR measure for T is contained in the Sobolev space W^{s}(S^{1} ×ℝ) for almost all (C^{r}generic, at least) f.. |

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Masato Tsujii, Physical measures for partially hyperbolic surface endomorphisms, *Acta Mathematica*, 10.1007/BF02392516, 194, 1, 37-132, 2005.01. |

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Jérôme Buzzi, Olivier Sester, Masato Tsujii, Weakly expanding skew-products of quadratic maps, *Ergodic Theory and Dynamical Systems*, 10.1017/S0143385702001694, 23, 5, 1401-1414, 2003.10, We consider quadratic skew-products over angle-doubling of the circle and prove that they admit positive Lyapunov exponents almost everywhere and an absolutely continuous invariant probability measure. This extends corresponding results of M. Viana and J. F. Alvès for skew-products over the linear strongly expanding map of the circle.. |

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Tsujii Masato, Rotation number and one-parameter Families of circle diffeo morphisms., *Ergodic theory and Dynamical systems*, 10.1017/S0143385700006805, 12, 2, 359-363, 1992.01, We consider one-parameter families of circle diffeomorphisms, f1(x) = f(x) + t(t C0(X, X) and A X a minimal set of f. We first introduce a new topological invariant, the D-function of a minimal set, by the investigation of the decomposition of the minimal set A under the action of fn n N. Then important properties about the invariant and the existence of minimal set with a given D-function in some subshift of finite type are discussed. Finally Sharkovskii's theorem is generalized to minimal sets of continuous mappings from the interval into itself.. |

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Masato Tsujii, Regular points for ergodic sinai measures, *Transactions of the American Mathematical Society*, 10.1090/S0002-9947-1991-1072103-1, 328, 2, 747-766, 1991.12, Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure μ if it is generic for μ and the Lyapunov exponents at it coincide with those of μ. We show that an ergodic measure with no zero Lyapunov exponent is absolutely continuous with respect to unstable foliation [L] if and only if the set of all points which are regular for it has positive Lebesgue measure.. |