Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1017/etds.2016.113

Language：English   Publishing type：Research paper (scientific journal)  

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。

DOI： 10.2969/jmsj/07027595

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1007/s00222-016-0701-5

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

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 - V₀)) where (.) denotes the spatial average on M. The number of the eigenvalues in the outermost annulus satisfies a Weyl law, that is, N^dVol (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.

Language：English  

Language：English   Publishing type：Research paper (scientific journal)  

If X is a contact Anosov vector field on a smooth compact manifold M and V∈C∞(M), it is known that the differential operator A=-X+V has some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev spaces. We show that for |Imz|→∞ the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band, the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate to the vertical line Re(z)=*D〉M, the space average of the function D(x)=V(x)-12divX|Eu(x) where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions. Si X est un champ de vecteur d'Anosov de contact sur une variété compacte lisse M et si V∈C∞(M), il est connu que l'opérateur différentiel A=-X+V a un spectre discret appelé résonances de Ruelle-Pollicott dans des espaces de Sobolev spécifiques. On montre que, pour |Imz|→∞, les valeurs propres de A sont incluses dans des bandes verticales et que, dans les gaps entre ces bandes, la résolvante de A est bornée uniformément par rapport à |Im(z)|. Dans chaque bande isolée, la densité des valeurs propres est donnée par une loi de Weyl. Dans la première bande, la plupart des valeurs propres se concentrent sur la ligne verticale Re(z)=*D〉M, qui est la moyenne spatiale de la fonction D(x)=V(x)-12divX|Eu(x), où Eu est la distribution instable. Ce spectre en bande permet d'exprimer le comportement asymptotique des fonctions de corrélations dynamiques. © 2013 .

DOI： 10.1016/j.crma.2013.04.022

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： http://dx.doi.org/10.1017/S0143385711000605

Language：English   Publishing type：Research paper (scientific journal)  

Language：English  

Language：English  

Publisher：Communications of the American Mathematical Society  

We develop a geometric micro-local analysis of contact Anosov flow, such as the geodesic flow on negatively curved manifold. This micro-local analysis is based on the wavepacket transform discussed by Faure and Tsujii [Ann. H. Lebesgue 6 (2023), pp. 331–426]. The main result is that the transfer operator is well approximated (in the high frequency limit) by the quantization of the Hamiltonian flow naturally defined from the contact Anosov flow and extended to some vector bundle over the symplecti- zation set. This has a few important consequences: the discrete eigenvalues of the gen­erator of transfer operators, called Ruelle spectrum, are structured into vertical bands. If the right-most band is isolated from the others, most of the Ruelle spectrum in it con­centrates along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tends to infinity. Some of these results were announced by Faure and Tsujii [C. R. Math. Acad. Sci. Paris 351 (2013), pp. 385–391].

DOI： 10.1090/cams/40

Scopus

Publishing type：Research paper (scientific journal)   Publisher：Kyushu Journal of Mathematics  

We introduce a class of discrete dynamical systems that we call virtually expanding. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron–Frobenius operator is quasi-compact on a Sobolev space of positive order for such a class of dynamical systems.

DOI： 10.2206/kyushujm.77.291

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Publishing type：Research paper (scientific journal)   Publisher：Annals of Mathematics  

We show that a topologically mixing C∞ Anosov ow on a 3-dimensional compact manifold is exponentially mixing with respect to any equilibrium measure with Holder potential

DOI： 10.4007/annals.2023.197.1.2

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Publisher：Annales Henri Lebesgue  

On a closed manifold M, we consider a smooth vector field X that generates an Anosov flow. Let V ∈ C∞(M; R) be a smooth function called potential. It is known that for any C>0, there exists some anisotropic Sobolev space HC such that the operator A=−X +V has intrinsic discrete spectrum on Re(z)>−C called Ruelle resonances. In this paper, we show a “Fractal Weyl law”: the density of resonances is bounded by O(⟨ω⟩ n 1+β0) where ω=Im(z), n=dim M−1 and 0<β0 ⩽ 1 is the Hölder exponent of the distribution Eu ⊕ Es (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances and the invertibility of the transfer operator. Since the dynamical distributions Eu, Es are non smooth, we use some semi-classical analysis based on wave packet transform associated to an adapted metric g on T ∗M and construct some specific anisotropic Sobolev spaces.

DOI： 10.5802/ahl.167

Scopus

Language：Others   Publishing type：Research paper (scientific journal)   Publisher：Elsevier {BV}  

We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.

DOI： 10.1016/j.jmaa.2022.126163

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Language：English   Publishing type：Research paper (scientific journal)  

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 : S¹→S¹. 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.

DOI： 10.1088/0951-7715/29/7/1917

Language：English   Publishing type：Research paper (scientific journal)  

For the Fourier transform mu of a general ( non- trivial) self- similar measure mu on the real line R, we prove a large deviation estimate

DOI： 10.1080/14689367.2015.1078291

Language：English  

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.

DOI： 10.1007/978-3-319-04807-9_2

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1017/S0143385711000605

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1088/0951-7715/23/7/001

Language：English   Publishing type：Research paper (scientific journal)  

We consider suspension semi-flows of angle-multiplying maps on the circle for C^r ceiling functions with r3. Under a C^rgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L² 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.

DOI： 10.1017/S0143385707000430

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.5802/aif.2253

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1088/0951-7715/19/10/011

Language：English   Publishing type：Research paper (scientific journal)  

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-.S¹ X ℝ-.S¹ × ℝ, T(x,y) = (ℓx, λy +f /(x)) where ≥ 2, 0 < λ < 1 and f is a C^r function on S¹. 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(S¹ ×ℝ) for almost all (C^rgeneric, at least) f.

DOI： 10.3934/dcds.2006.15.21

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/BF02392516

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1017/S0143385702001694

Language：English   Publishing type：Research paper (scientific journal)  

A measure on the space of smooth mappings and dynamical system theory

DOI： 10.2969/jmsj/04430415

Language：English  

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.

DOI： 10.1017/S0143385700006805

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1090/S0002-9947-1991-1072103-1

Event date： 2019.3

Language：English   Presentation type：Oral presentation (general)  

Venue：慶應義塾大学   Country：Japan  

Event date： 2018.8

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Federal university of Salvador   Country：Brazil  

Event date： 2017.12 - 2016.12

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：広島大学   Country：Japan  

Event date： 2017.12 - 2016.12

Language：English   Presentation type：Oral presentation (general)  

Venue：Aussois   Country：France  

Event date： 2017.7 - 2016.7

Language：English   Presentation type：Oral presentation (general)  

Venue：The university of Tokyo   Country：France  

Event date： 2017.1 - 2016.1

Language：English   Presentation type：Oral presentation (general)  

Venue：The university of Tokyo   Country：Japan  

Event date： 2016.6 - 2016.7

Language：English   Presentation type：Oral presentation (general)  

Venue：The Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy   Country：Japan  

Event date： 2016.4 - 2016.5

Language：English   Presentation type：Oral presentation (general)  

Venue：Erwin Schrodinger Institute, Vienna, Austria   Country：Japan  

Other Link： http://www.esi.ac.at/activities/events/2016/mixing-flows-and-averaging-methods

Event date： 2016.3

Language：English  

Venue：ICERM, Brown university   Country：Japan  

Other Link： https://icerm.brown.edu/programs/sp-s16/w2/

Event date： 2015.7 - 2015.8

Language：English   Presentation type：Oral presentation (general)  

Venue：ICTP, Trieste, Italy   Country：Italy  

Other Link： http://indico.ictp.it/event/a14293/

Event date： 2014.8

Language：English   Presentation type：Oral presentation (general)  

Venue：Coex, Seoul   Country：Korea, Republic of  

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.

Event date： 2014.8

Language：English   Presentation type：Oral presentation (general)  

Venue：忠南大学, Seoul   Country：Korea, Republic of  

Event date： 2013.12

Language：English   Presentation type：Oral presentation (general)  

Venue：慶応大学   Country：Japan  

We consider the semi-classical (or Gutzwiller-Voros) zeta functions for &#36;C^infty&#36; contact Anosov flows. Analyzing the spectra of the generators for some transfer operators associated to the flow, we prove, for any &#36; au>0&#36;, that its zeros are contained in the union of the &#36; au&#36;-neighborhood of the imaginary axis, &#36;|Re(s)|< au&#36;, and the region &#36;Re(s)<-chi_0+ au&#36;, up to finitely many exceptions, where &#36;chi_0>0&#36; 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.

Other Link： http://www.math.keio.ac.jp/~nakada/dynamical_systems_2013-dec/program.pdf

Event date： 2013.12

Language：English   Presentation type：Oral presentation (general)  

Venue：CIRM, Marseille   Country：France  

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.

Other Link： http://hasselblatttroubetzkoy.weebly.com/information1.html

Event date： 2012.6

Language：English   Presentation type：Oral presentation (general)  

Venue：ICTP, Trieste   Country：Italy  

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.

Event date： 2011.3

Language：Others   Presentation type：Oral presentation (general)  

Venue：京都大学   Country：Japan  

Event date： 2011.3

Language：Others   Presentation type：Oral presentation (general)  

Venue：京都大学   Country：Japan  

Event date： 2011.1

Language：Others   Presentation type：Oral presentation (general)  

Venue：東京工業大学   Country：Japan  

Event date： 2011.1 - 2011.7

Language：Others   Presentation type：Oral presentation (general)  

Venue：Mittag-Leffeler Institute   Country：Sweden  

Event date： 2011.1

Language：Others   Presentation type：Oral presentation (general)  

Venue：東京工業大学   Country：Japan  

Event date： 2010.5 - 2010.6

Language：Others   Presentation type：Oral presentation (general)  

Venue：Corinaldo   Country：Italy  

Event date： 2009.11

Language：Others   Presentation type：Oral presentation (general)  

Venue：Oberwalfach 数学研究所   Country：Germany  

Event date： 2009.6

Language：Others   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：rwin Schrodinger Institute, Vienna   Country：Austria  

Event date： 2008.5

Language：Others   Presentation type：Oral presentation (general)  

Venue：NCTS, Shinchu   Country：Taiwan, Province of China  

Event date： 2008.2 - 2008.3

Language：Others   Presentation type：Oral presentation (general)  

Venue：CIRM, Marseille   Country：France  

Language：Others   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：中央大学   Country：Japan  

Language：Others   Presentation type：Oral presentation (general)  

Venue：広島大学理学部   Country：Japan  

Event date： 2023.3

Language：English   Presentation type：Oral presentation (general)  

Venue：北海道大学   Country：Japan  

Language：Others   Presentation type：Oral presentation (general)  

Venue：九州大学理学部   Country：Japan  

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：4

Number of peer-reviewed articles in Japanese journals：0

Proceedings of International Conference Number of peer-reviewed papers：0

Proceedings of domestic conference Number of peer-reviewed papers：0

Type：Academic society, research group, etc. 

Type：Competition, symposium, etc. 

Number of participants：10

Type：Competition, symposium, etc. 

Number of participants：50

Type：Academic society, research group, etc. 

Type：Academic society, research group, etc. 

Type：Academic society, research group, etc. 

Type：Peer review 

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Type：Peer review 

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Type：Seminar, workshop

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Audience：General,　Scientific,　Company,　Civic organization,　Governmental agency

Type：Seminar, workshop

エクセレント・スチューデント・イン・サイエンス事業は科学技術振興機構と九州大学が行っている意欲・能力ある高校生を次世代の科学者に育てるための取り組みである．この講座において数学部門の担当をして，半年間におよび月二回のセミナーを行った．

Type：Seminar, workshop

エクセレント・スチューデント・イン・サイエンス事業は科学技術振興機構と九州大学が行っている意欲・能力ある高校生を次世代の科学者に育てるための取り組みである．この講座において数学部門の担当をして，半年間におよび月二回のセミナーを行った．

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Audience：General,　Scientific,　Company,　Civic organization,　Governmental agency

Type：Lecture

Type：Visiting lecture

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