記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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.

記述言語：英語  

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

記述言語：英語  

記述言語：英語  

出版者・発行元：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

掲載種別：研究論文（学術雑誌）   出版者・発行元：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

Scopus

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掲載種別：研究論文（学術雑誌）   出版者・発行元：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

Scopus

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出版者・発行元：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

記述言語：その他   掲載種別：研究論文（学術雑誌）   出版者・発行元：Journal of Mathematical Analysis and Applications  

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

Scopus

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記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

DOI： 10.1007/BF02392516

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

DOI： 10.2969/jmsj/04430415

記述言語：英語  

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

記述言語：英語   掲載種別：研究論文（学術雑誌）  

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

開催年月日： 2019年3月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：慶應義塾大学   国名：日本国  

開催年月日： 2018年8月

記述言語：英語   会議種別：口頭発表（招待・特別）  

開催地：Federal university of Salvador   国名：ブラジル連邦共和国  

開催年月日： 2017年12月 - 2016年12月

記述言語：英語   会議種別：口頭発表（招待・特別）  

開催地：広島大学   国名：日本国  

開催年月日： 2017年12月 - 2016年12月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：Aussois   国名：フランス共和国  

開催年月日： 2017年7月 - 2016年7月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：The university of Tokyo   国名：フランス共和国  

開催年月日： 2017年1月 - 2016年1月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：The university of Tokyo   国名：日本国  

開催年月日： 2016年6月 - 2016年7月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：The Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy   国名：日本国  

開催年月日： 2016年4月 - 2016年5月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：Erwin Schrodinger Institute, Vienna, Austria   国名：日本国  

その他リンク： http://www.esi.ac.at/activities/events/2016/mixing-flows-and-averaging-methods

開催年月日： 2016年3月

記述言語：英語  

開催地：ICERM, Brown university   国名：日本国  

その他リンク： https://icerm.brown.edu/programs/sp-s16/w2/

開催年月日： 2015年7月 - 2015年8月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：ICTP, Trieste, Italy   国名：イタリア共和国  

その他リンク： http://indico.ictp.it/event/a14293/

開催年月日： 2014年8月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：Coex, Seoul   国名：大韓民国  

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.

開催年月日： 2014年8月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：忠南大学, Seoul   国名：大韓民国  

開催年月日： 2013年12月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：慶応大学   国名：日本国  

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.

その他リンク： http://www.math.keio.ac.jp/~nakada/dynamical_systems_2013-dec/program.pdf

開催年月日： 2013年12月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：CIRM, Marseille   国名：フランス共和国  

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.

その他リンク： http://hasselblatttroubetzkoy.weebly.com/information1.html

開催年月日： 2012年6月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：ICTP, Trieste   国名：イタリア共和国  

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.

開催年月日： 2011年3月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：京都大学   国名：日本国  

開催年月日： 2011年3月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：京都大学   国名：日本国  

開催年月日： 2011年1月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：東京工業大学   国名：日本国  

開催年月日： 2011年1月 - 2011年7月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：Mittag-Leffeler Institute   国名：スウェーデン王国  

開催年月日： 2011年1月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：東京工業大学   国名：日本国  

開催年月日： 2010年5月 - 2010年6月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：Corinaldo   国名：イタリア共和国  

開催年月日： 2009年11月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：Oberwalfach 数学研究所   国名：ドイツ連邦共和国  

開催年月日： 2009年6月

記述言語：その他   会議種別：公開講演，セミナー，チュートリアル，講習，講義等  

開催地：rwin Schrodinger Institute, Vienna   国名：オーストリア共和国  

開催年月日： 2008年5月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：NCTS, Shinchu   国名：台湾  

開催年月日： 2008年2月 - 2008年3月

記述言語：その他   会議種別：口頭発表（一般）  

開催地：CIRM, Marseille   国名：フランス共和国  

記述言語：その他   会議種別：口頭発表（一般）  

開催地：広島大学理学部   国名：日本国  

記述言語：その他   会議種別：公開講演，セミナー，チュートリアル，講習，講義等  

開催地：中央大学   国名：日本国  

開催年月日： 2023年3月

記述言語：英語   会議種別：口頭発表（一般）  

開催地：北海道大学   国名：日本国  

記述言語：その他   会議種別：口頭発表（一般）  

開催地：九州大学理学部   国名：日本国  

種別：査読等 

外国語雑誌　査読論文数：4

日本語雑誌　査読論文数：0

国際会議録　査読論文数：0

国内会議録　査読論文数：0

種別：学会・研究会等 

種別：大会・シンポジウム等 

参加者数：10

種別：大会・シンポジウム等 

参加者数：50

種別：学会・研究会等 

種別：学会・研究会等 

種別：学会・研究会等 

種別：査読等 

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種別：査読等 

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対象：社会人・一般,　学術団体,　企業,　市民団体,　行政機関

種別：セミナー・ワークショップ

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．

種別：セミナー・ワークショップ

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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対象：社会人・一般,　学術団体,　企業,　市民団体,　行政機関

種別：講演会

種別：出前授業

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