2003 Fiscal Year Annual Research Report

撞球系をモデルとした力学系・カオス・フラクタルの大域解析的研究

Research Project

Project/Area Number	13440056
Research Institution	Hiroshima Institute of Technology
Principal Investigator	久保泉広島工業大学, 環境学部, 教授 (70022621)
Co-Investigator(Kenkyū-buntansha)	印南信宏新潟大学, 理学部, 教授 (20160145) 由利美智子札幌大学, 経営学部, 教授 (70174836) 盛田健彦広島大学, 大学院・理学研究科, 教授 (00192782) 宍倉光広京都大学, 大学院・理学研究科, 教授 (70192606) 田崎秀一早稲田大学, 理工学部, 教授 (10260150)
Keywords	撞球問題 / 離散力学系 / カオス / フラクタル / 記号力学系 / 古典力学系 / 1次元力学系 / 複素力学系
Research Abstract	蝕無しの条件をみたす2次元散乱撞球にリプシッツ連続な不変葉層構造が存在することを証明していたが、このリプシッツ連続性のより構成的で、初等的な証明を新たに得たことにより、更なる進展が期待される。 Busemannによって造られた測地線の幾何学的手法を平面凸ビリヤードの配位空間の幾何に持ち込み、平行線の理論を展開することによりいくつかの新しい事実を見出した。また、曲線\|x/a\|^n+\|y/b\|^n=1で囲まれた領域の平面凸ビリヤードのシミュレーションで興味ある現象を見出した。この撞球系のポアンカレ写像を調べることにより、定性的な性質が明らかになり、シミュレーションで観察される現象を理論的に解明する手懸りが得られた。また、当課題の研究過程で谷口の力学系を用いた乱数生成法に興味を抱き、解析方法を得た。次年度以降の課題に取り上げたい。周期的古典ビリヤード系を単純化した写像において輸送現象が統計分布の時間発展演算子のスペクトル及び固有関数で特徴付けられることを示し、量子ビリヤード系における境界要素法と散乱問題の関係を調べた。複素力学系の研究において、放物型不動点に関連した場合にのみ知られていたくりこみで不変な写像の族について、その摂動でも不変になる族が構成できた。無限対1の区分的に可逆なマルコフ系について、弱ギブス測度に関わる大偏差原理を調べ、レベル2の大偏差とある条件下での指数減少を示した。さらに、大偏差原理の多重フラクタル版を考察した。その他、Boyle-Maassの予想の証明を得た。間欠カオスが見られる位置依存型ランダム力学系の具体例について不変測度の評価を行った。量子カオスと関連の深いランダム行列の極限分布の挙動を、自由確率論の立場から研究した。また複素構造のある族の臨界点において、再帰性を保存するような変形の自由度を解析した。直交多項式の母関数のウイナーカオスの視点から調べた。

Research Products

(50 results)

All Other

All Publications (50 results)

[Publications] Nobuhiro Asai: "Multiplicative renormalization and generating functions II"Taiwanese Journal of Mathematics. 8・4. (2004)
[Publications] J.-Y, Briend: "Linearity of the exceptional set for maps"Math.Ann.2004. (2004)
[Publications] Tomoki Inoue: "Ergodic sums of non-integrable functions under one-dimensional dynamical systems with indifferent fixed"Ergodic Theory and Dynamical Systems. (2004)
[Publications] Shuichi Tasaki: "Control of Decoherence : dynamical decoupling vs quantum Zeno effect"International Journal of Quantum Chemistry. (2004)
[Publications] Shuichi Tasaki: "On entropy production of a one-dimensional lattice conductor"Quantum Information V. (2004)
[Publications] Shuichi Tasaki: "On Prigogine's Approaches to Irreversibility - A case study by the baker map -"Discrete Dynamics in Nature and Society. (2004)
[Publications] Izumi Kubo: "Generating functions of exponential type for orthogonal polynomials"Infinite Dimensional Analysis, Quantum Probability and Related Topics. 7・1. 155-159 (2004)
[Publications] Michiko Yuri: "Thermodynamic Formalism for countable to one Markov System"Transactions of the American Mathematical Society. 355・7. 2949-2917 (2003)
[Publications] Michiko Yuri: "Weak Gibbs measares for Internittent Systems and Weakly Gibbsian States in Statis tical Mechanics"Communications in Mathematical Physics. 241. 453-466 (2003)
[Publications] Takehiko Morita: "Piecewise $C^2$ perturbation of Lasota-Yorke maps and their ergodic properties"Osaka J.Math.. 40. 207-233 (2003)
[Publications] Nobuhiro Asai: "Gaussian and Poisson white noises with related characterization theorems"Contemporary Mathematics, American Mathematical Society. 317. 45-56 (2003)
[Publications] Nobuhiro Asai: "Multiplicative renormalization and generating functions I"Taiwanese Journal of Mathematics. 7・1. 89-101 (2003)
[Publications] Satoshi Ikeda: "Impact of Local Topological Information on Random Walks on Finite Graphs"Thirtieth International Colloquium on Automata, Languages and Programming2. 203-207 (2003)
[Publications] Satoshi Ikeda: "Reducing the Hitting and the Cover Times of Random Walk on Finite Graph by Local Topological Information"Proceedings of The 2003 International Conference on VLSI. 203-207 (2003)
[Publications] Nobuhiro Asai: "Generating functions of orhtogonal polynomials and Szeg\"o-Jacobi parameters"Prob.Math.Stat.. 27-291 (2003)
[Publications] Yoshimichi Ueda: "Fullness, Connes' $\chi$-groups, and ultra-products of amalgamated free products over Cartan subalgebras"Trans Amer.Math.Soc.. 355. 349-371 (2003)
[Publications] Fumio Hiai: "Automorphisms of free product type and their crossed-products"J.Operator Theory. 50・1. 119-130 (2003)
[Publications] Yoshimichi Ueda: "Free product actions and their applications"Quantum Probability and White Noise Analysis. 16. 388-411 (2003)
[Publications] Shuichi Tasaki: "Spectral characterization of anomalous diffusion of a periodic piecewise linear intermittent map"Physica. D 183. 205-219 (2003)
[Publications] H.Makino: "Non-Poissonian Level Spacing of Classically Integrable Quantum Systems Based on the Berry-Robnik Approach"Progress of Theoretical Physics, Supplement. 150. 376-380 (2003)
[Publications] H.Makino: "Derivation from Berry-Robnik distribution caused by spectral accumulation"Journal of Physical Society of Japan. 72-C. 97-100 (2003)
[Publications] M.Fujiyoshi: Friedrichs model with singular continuous spectrum. 72-C. 73-76 (2003)
[Publications] Y.Okada: "Can one determine the shape of a quantum billiard table through the eigenenergies and resonances?"Progress of Theoretical Physics, Supplement. 150. 397-400 (2003)
[Publications] Shuichi Tasaki: "Fluctuation Theorem, Nonequilibrium Steady States and MacLennan-Zubarev Ensembles of a Class of Large Quantum Systems"Fundamental Aspects of Quantum Physics. 100-119 (2003)
[Publications] H.Makino: "Level spacing statistics of classically integrable systems : Investigation along the lines of the Berry-Robnik approach"Physical Review. E67・06. 6205 (2003)
[Publications] M.Pollicott: "Zeta Functions for Certain Multi-Dimensional Non-Hyperbolic Maps"Nonlinearity. 14・5. 1265-1278 (2002)
[Publications] Michiko Yuri: "On the Speed of Convergence to Equilibrium States for Multi-Dimensional Maps with Indifferent Periodic Points"Nonlinearity. 15・2. 429-445 (2002)
[Publications] Michiko Yuri: "Multifractal Analysis of Weak Gibbs Measures for Intermittent Systems"Communications in Mathematical Physics. 230・2. 365-388 (2002)
[Publications] Michiko Yuri: "Weak Gibbs measures and the local product structure"Ergodic Theory and Dynamical Systems. 22. 1933-1955 (2002)
[Publications] Nobuhiro Innami: "Geometry of geodesics for convex billiards and circular billiards"Nihonkai Math.J.. 13・1. 73-120 (2002)
[Publications] Mitsuhiro Shishikura: "Minimal sets of certain annular homeomorphisms"Hiroshima Math.J.. 32・2. 207-215 (2002)
[Publications] Masakazu Nasu: "The dynamics of expansive invertible onesided cellular automata"Trans.Amer.Math.Soc.. 354. 4067-4084 (2002)
[Publications] Satoshi Ikeda: "Fair Circulation of a Token"IEEE Transactions on Parallel and Distributed Systems. 13・4. 367-372 (2002)
[Publications] Nobuhiro Asai: "Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures"Proc.Amer.Math.Soc.. 131・3. 815-823 (2002)
[Publications] Shuichi Tasaki: "Thermodynamic Behaviors of Large Dynamical Systems - quantum 1d conductor and classical multibaker map-"in "Dynamics of Dissipation", Lecture Notes in Physics. 597. 395-412 (2002)
[Publications] Shuichi Tasaki: "Spectral Properties of a Piecewise Linear Intermittent Map"Journal of Statistical Physics. 109. 803-820 (2002)
[Publications] Shuichi Tasaki: "Irreversibility in Reversible Multibaker Maps"Advances in Chemical Physics. 122. 77-107 (2002)
[Publications] H.Honda: "Relaxation phenomenon measured as dynamical specific heat in first-order phase transition of molecular crystal"Physical Review B. 65. 149-166 (2002)
[Publications] D.Shlyakhtenko: "Irreducible subfactors of $L(F_\infty)$ of index $\lambda > 4$"J.reine angew.Math.. 149-166 (2002)
[Publications] M.Pollicott: "Statistical properties of maps with indifferent periodic points"COMMUNICATIONS IN MATHEMATICAL PHYSICS. 217・3. 503-520 (2002)
[Publications] Takehiko Morita: "Construction of K-stable foliations for two-dimensional dispersing Billiards without eclipse"J.Math.Soc.Japan. 56-3. 803-831 (2004)
[Publications] Nobuhiro Innami: "Jacobi vector fields along geodesics in glued Riemannian manifolds"Nihonkai Math.J.. 12・1. 101-112 (2001)
[Publications] Nobuhiro Innami: "Gradient vector fields which characterize warped products"Math.Scand. 88・2. 182-192 (2001)
[Publications] Nobuhiro Asahi: "General characterization theorems and intrinsic topologies in white noise analysis"Hiroshima Mathematical Journal. 31. 299-330 (2001)
[Publications] Nobuhiro Asahi: "Roles of log-concavity, log-convexity, and growth order in white noise analysis"Infinite Dimiensional Analysis, Quantum Probability and Related Topics. 4・1. 59-84 (2001)
[Publications] Shuichi Tasaki: "Current Fluctuations in Nonequilibrium Steady States for a One-Dimensional Lattice Conductor"Quantum Information III. 157-176 (2001)
[Publications] Shuichi Tasaki: "Nonequilibrium stationary states of noninteracting electrons in a one-dimensional lattice"Chaos, Solitons and Fractals. 122. 2657-2674 (2001)
[Publications] P.Gaspard: "Liouvillian dynamics of the Hopf bifurcation"Physical Review E. 6405・5. 6232 (2001)
[Publications] Tomoki Inoue: "Correction to 'Ratio ergodic theorems for maps with indifferent fixed points'"Ergod.Theory and Dynamical Systems.. 21. 1273-1273 (2001)
[Publications] Yoshimichi Ueda: "Remarks on free products with respect to non-tracial states"Math.Scand.. 88. 111-125 (2001)

2003 Fiscal Year Annual Research Report

撞球系をモデルとした力学系・カオス・フラクタルの大域解析的研究

Principal Investigator

久保 泉 広島工業大学, 環境学部, 教授 (70022621)

Research Products

[Publications] Nobuhiro Asai: "Multiplicative renormalization and generating functions II"Taiwanese Journal of Mathematics. 8・4. (2004)

[Publications] J.-Y, Briend: "Linearity of the exceptional set for maps"Math.Ann.2004. (2004)

[Publications] Tomoki Inoue: "Ergodic sums of non-integrable functions under one-dimensional dynamical systems with indifferent fixed"Ergodic Theory and Dynamical Systems. (2004)

[Publications] Shuichi Tasaki: "Control of Decoherence : dynamical decoupling vs quantum Zeno effect"International Journal of Quantum Chemistry. (2004)

[Publications] Shuichi Tasaki: "On entropy production of a one-dimensional lattice conductor"Quantum Information V. (2004)

[Publications] Shuichi Tasaki: "On Prigogine's Approaches to Irreversibility - A case study by the baker map -"Discrete Dynamics in Nature and Society. (2004)

[Publications] Izumi Kubo: "Generating functions of exponential type for orthogonal polynomials"Infinite Dimensional Analysis, Quantum Probability and Related Topics. 7・1. 155-159 (2004)

[Publications] Michiko Yuri: "Thermodynamic Formalism for countable to one Markov System"Transactions of the American Mathematical Society. 355・7. 2949-2917 (2003)

[Publications] Michiko Yuri: "Weak Gibbs measares for Internittent Systems and Weakly Gibbsian States in Statis tical Mechanics"Communications in Mathematical Physics. 241. 453-466 (2003)

[Publications] Takehiko Morita: "Piecewise $C^2$ perturbation of Lasota-Yorke maps and their ergodic properties"Osaka J.Math.. 40. 207-233 (2003)

[Publications] Nobuhiro Asai: "Gaussian and Poisson white noises with related characterization theorems"Contemporary Mathematics, American Mathematical Society. 317. 45-56 (2003)

[Publications] Nobuhiro Asai: "Multiplicative renormalization and generating functions I"Taiwanese Journal of Mathematics. 7・1. 89-101 (2003)

[Publications] Satoshi Ikeda: "Impact of Local Topological Information on Random Walks on Finite Graphs"Thirtieth International Colloquium on Automata, Languages and Programming2. 203-207 (2003)

[Publications] Satoshi Ikeda: "Reducing the Hitting and the Cover Times of Random Walk on Finite Graph by Local Topological Information"Proceedings of The 2003 International Conference on VLSI. 203-207 (2003)

[Publications] Nobuhiro Asai: "Generating functions of orhtogonal polynomials and Szeg\"o-Jacobi parameters"Prob.Math.Stat.. 27-291 (2003)

[Publications] Yoshimichi Ueda: "Fullness, Connes' $\chi$-groups, and ultra-products of amalgamated free products over Cartan subalgebras"Trans Amer.Math.Soc.. 355. 349-371 (2003)

[Publications] Fumio Hiai: "Automorphisms of free product type and their crossed-products"J.Operator Theory. 50・1. 119-130 (2003)

[Publications] Yoshimichi Ueda: "Free product actions and their applications"Quantum Probability and White Noise Analysis. 16. 388-411 (2003)

[Publications] Shuichi Tasaki: "Spectral characterization of anomalous diffusion of a periodic piecewise linear intermittent map"Physica. D 183. 205-219 (2003)

[Publications] H.Makino: "Non-Poissonian Level Spacing of Classically Integrable Quantum Systems Based on the Berry-Robnik Approach"Progress of Theoretical Physics, Supplement. 150. 376-380 (2003)

[Publications] H.Makino: "Derivation from Berry-Robnik distribution caused by spectral accumulation"Journal of Physical Society of Japan. 72-C. 97-100 (2003)

[Publications] M.Fujiyoshi: Friedrichs model with singular continuous spectrum. 72-C. 73-76 (2003)

[Publications] Y.Okada: "Can one determine the shape of a quantum billiard table through the eigenenergies and resonances?"Progress of Theoretical Physics, Supplement. 150. 397-400 (2003)

[Publications] Shuichi Tasaki: "Fluctuation Theorem, Nonequilibrium Steady States and MacLennan-Zubarev Ensembles of a Class of Large Quantum Systems"Fundamental Aspects of Quantum Physics. 100-119 (2003)

[Publications] H.Makino: "Level spacing statistics of classically integrable systems : Investigation along the lines of the Berry-Robnik approach"Physical Review. E67・06. 6205 (2003)

[Publications] M.Pollicott: "Zeta Functions for Certain Multi-Dimensional Non-Hyperbolic Maps"Nonlinearity. 14・5. 1265-1278 (2002)

[Publications] Michiko Yuri: "On the Speed of Convergence to Equilibrium States for Multi-Dimensional Maps with Indifferent Periodic Points"Nonlinearity. 15・2. 429-445 (2002)

[Publications] Michiko Yuri: "Multifractal Analysis of Weak Gibbs Measures for Intermittent Systems"Communications in Mathematical Physics. 230・2. 365-388 (2002)

[Publications] Michiko Yuri: "Weak Gibbs measures and the local product structure"Ergodic Theory and Dynamical Systems. 22. 1933-1955 (2002)

[Publications] Nobuhiro Innami: "Geometry of geodesics for convex billiards and circular billiards"Nihonkai Math.J.. 13・1. 73-120 (2002)

[Publications] Mitsuhiro Shishikura: "Minimal sets of certain annular homeomorphisms"Hiroshima Math.J.. 32・2. 207-215 (2002)

[Publications] Masakazu Nasu: "The dynamics of expansive invertible onesided cellular automata"Trans.Amer.Math.Soc.. 354. 4067-4084 (2002)

[Publications] Satoshi Ikeda: "Fair Circulation of a Token"IEEE Transactions on Parallel and Distributed Systems. 13・4. 367-372 (2002)

[Publications] Nobuhiro Asai: "Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures"Proc.Amer.Math.Soc.. 131・3. 815-823 (2002)

[Publications] Shuichi Tasaki: "Thermodynamic Behaviors of Large Dynamical Systems - quantum 1d conductor and classical multibaker map-"in "Dynamics of Dissipation", Lecture Notes in Physics. 597. 395-412 (2002)

[Publications] Shuichi Tasaki: "Spectral Properties of a Piecewise Linear Intermittent Map"Journal of Statistical Physics. 109. 803-820 (2002)

[Publications] Shuichi Tasaki: "Irreversibility in Reversible Multibaker Maps"Advances in Chemical Physics. 122. 77-107 (2002)

[Publications] H.Honda: "Relaxation phenomenon measured as dynamical specific heat in first-order phase transition of molecular crystal"Physical Review B. 65. 149-166 (2002)

[Publications] D.Shlyakhtenko: "Irreducible subfactors of $L(F_\infty)$ of index $\lambda > 4$"J.reine angew.Math.. 149-166 (2002)

[Publications] M.Pollicott: "Statistical properties of maps with indifferent periodic points"COMMUNICATIONS IN MATHEMATICAL PHYSICS. 217・3. 503-520 (2002)

[Publications] Takehiko Morita: "Construction of K-stable foliations for two-dimensional dispersing Billiards without eclipse"J.Math.Soc.Japan. 56-3. 803-831 (2004)

[Publications] Nobuhiro Innami: "Jacobi vector fields along geodesics in glued Riemannian manifolds"Nihonkai Math.J.. 12・1. 101-112 (2001)

[Publications] Nobuhiro Innami: "Gradient vector fields which characterize warped products"Math.Scand. 88・2. 182-192 (2001)

[Publications] Nobuhiro Asahi: "General characterization theorems and intrinsic topologies in white noise analysis"Hiroshima Mathematical Journal. 31. 299-330 (2001)

[Publications] Nobuhiro Asahi: "Roles of log-concavity, log-convexity, and growth order in white noise analysis"Infinite Dimiensional Analysis, Quantum Probability and Related Topics. 4・1. 59-84 (2001)

[Publications] Shuichi Tasaki: "Current Fluctuations in Nonequilibrium Steady States for a One-Dimensional Lattice Conductor"Quantum Information III. 157-176 (2001)

[Publications] Shuichi Tasaki: "Nonequilibrium stationary states of noninteracting electrons in a one-dimensional lattice"Chaos, Solitons and Fractals. 122. 2657-2674 (2001)

[Publications] P.Gaspard: "Liouvillian dynamics of the Hopf bifurcation"Physical Review E. 6405・5. 6232 (2001)

[Publications] Tomoki Inoue: "Correction to 'Ratio ergodic theorems for maps with indifferent fixed points'"Ergod.Theory and Dynamical Systems.. 21. 1273-1273 (2001)

[Publications] Yoshimichi Ueda: "Remarks on free products with respect to non-tracial states"Math.Scand.. 88. 111-125 (2001)

久保泉広島工業大学, 環境学部, 教授 (70022621)