2003 年度実績報告書

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

研究課題

研究課題/領域番号	13440056
研究機関	広島工業大学
研究代表者	久保泉広島工業大学, 環境学部, 教授 (70022621)
研究分担者	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)
キーワード	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
研究概要	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。

研究成果
(50件)

すべてその他

すべて文献書誌 (50件)

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

2003 年度 実績報告書

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

研究代表者

久保 泉 広島工業大学, 環境学部, 教授 (70022621)

研究成果

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

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

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

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

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

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

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

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

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

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

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

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

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

[文献書誌] 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)

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

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

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

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

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

[文献書誌] 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)

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

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

[文献書誌] 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)

[文献書誌] 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)

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

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

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

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

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

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

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

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

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

[文献書誌] 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)

[文献書誌] 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)

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

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

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

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

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

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

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

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

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

[文献書誌] 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)

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

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

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

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

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

2003 年度実績報告書

久保泉広島工業大学, 環境学部, 教授 (70022621)