Topics in Smooth Ergodic Theory: Stochastic Properties, Thermodynamic Formalism, Coexistence

平滑遍历理论主题:随机性质、热力学形式主义、共存

基本信息

  • 批准号:
    2153053
  • 负责人:
  • 金额:
    $ 22.43万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2022
  • 资助国家:
    美国
  • 起止时间:
    2022-06-01 至 2025-05-31
  • 项目状态:
    未结题

项目摘要

One of the greatest discoveries of the second half of the last century, which impacted many branches of science, was the phenomenon known as ``deterministic chaos'' – the emergence of irregular chaotic motions in purely deterministic systems. Models with this type of motions can be widely found in physics, biology, chemistry, as well as in engineering and economics. Hyperbolicity theory, which is an important part of general theory of dynamical systems, provides a mathematical foundation for the deterministic chaos phenomenon by supplying researchers with tools that allow them to describe global properties of a nonlinear dynamical system using information about infinitesimal hyperbolic behavior of its trajectories. The study of hyperbolic phenomena originated in the seminal works of Artin, Morse, Hedlund and Hopf, but the systematic study of hyperbolic dynamical systems was initiated by Smale, Anosov and Sinai, who studied dynamical systems with strong hyperbolic behavior which possess high level of unpredictability and exhibit strong chaotic behavior. In this project, the PI considers the weakest (hence, most general) form of hyperbolicity, known as non-uniform hyperbolicity. The latter originated in the work of the PI. The study of non-uniformly hyperbolic systems is based upon the theory of Lyapunov exponents, which provides some “practical” tools to detect and describe hyperbolic properties of the systems. The modern non-uniform hyperbolicity theory has numerous applications to ergodic theory, mathematical and statistical physics, Riemannian geometry, and other areas of mathematics and beyond. The project provides research training opportunities for graduate students.Building upon past results the PI will carry out a broad research program which includes the following topics: (1) Thermodynamic formalism for non-uniformly hyperbolic dynamical systems. Some ideas from geometric measure theory are used to construct equilibrium measures in hyperbolic dynamics. The new method is based on pushing forward by the dynamics the Caratheodory measure associated with the Caratheodory dimension structure generated by the potential; (2) Essential coexistence of hyperbolic and non-hyperbolic behavior. This is to understand how two different types of dynamical behavior - fully hyperbolic (positive entropy) and non-hyperbolic (zero entropy) - can coexist in an essential way. The project is aimed at constructing Hamiltonian systems and geodesic flows which exhibit the essential coexistence phenomenon thus providing new insights in the classical Kolmogorov-Arnold-Moser theory; (3) The study of two important conjectures in dynamics: (i) Katok's entropy conjecture, claiming that a volume preserving uniquely ergodic diffeomorphism has zero Kolmogorov-Sinai entropy; (ii) Baire Category conjecture, claiming that irregular sets for smooth dynamical systems and for continuous cocycles over them have the 2nd Baire Category; (4) Maps with exponential and polynomial decay of correlations on compact manifolds. The goal is to substantially advance the understanding of smooth realization problem by showing that any smooth manifold admits a volume preserving hyperbolic diffeomorphism with polynomial or exponential decay of correlations and also satisfies the Central Limit Theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
上个世纪后半叶最伟大的发现之一,影响了许多科学分支,是被称为“确定性混沌”的现象--在纯粹确定性系统中出现不规则的混沌运动。具有这种类型的运动的模型可以广泛地在物理学、生物学、化学以及工程和经济学中找到。双曲性理论是动力系统一般理论的重要组成部分,它为研究人员提供了一种工具,使他们能够利用非线性动力系统轨迹的无穷小双曲行为的信息来描述系统的全局性质,从而为确定性混沌现象提供了数学基础。双曲现象的研究起源于Artin,莫尔斯,Hedlund和Hopf的开创性著作,但双曲动力系统的系统研究是由Smale,Anosov和Sinai发起的,他们研究具有强双曲行为的动力系统,这些动力系统具有高水平的不可预测性并表现出强混沌行为。在这个项目中,PI考虑了双曲性的最弱(因此也是最普遍的)形式,称为非均匀双曲性。后者起源于PI的工作。李雅普诺夫指数理论是研究非一致双曲系统的基础,它为检测和描述非一致双曲系统的双曲性提供了“实用”工具。现代非均匀双曲性理论在遍历理论、数学和统计物理、黎曼几何和其他数学领域以及更远的领域有着众多的应用。该项目为研究生提供了研究培训的机会。PI将在过去成果的基础上开展广泛的研究计划,其中包括以下主题:(1)非一致双曲动力系统的热力学形式。利用几何测度理论中的一些思想来构造双曲动力学的平衡测度。新方法是基于动力学推进与势产生的Caratheodory维数结构相关联的Caratheodory测度;(2)双曲与非双曲行为的本质共存。这是为了理解两种不同类型的动力学行为-全双曲(正熵)和非双曲(零熵)-如何以一种基本的方式共存。(3)研究动力学中的两个重要问题:(1)Katok熵猜想,即保体积的唯一遍历同态的Kolmogorov-Sinai熵为零;(ii)Baire范畴猜想,认为光滑动力系统的不规则集和其上的连续上圈都有第二Baire范畴;(4)紧流形上的关联指数衰减和多项式衰减的映射。其目标是通过证明任何光滑流形都允许具有多项式或指数衰减的相关性的保体积双曲型同构,并满足中心极限定理,从而大大推进对光滑实现问题的理解。该奖项反映了NSF的法定使命,并通过使用基金会的智力价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Yakov Pesin其他文献

明治三十年前半の帝国教育会における研究活動の展開 -学制調査部と国字改良部に注目して-
1900年代上半期帝国教育会研究活动的开展 -以教育制度研究部和国民素质提高部为中心-
明治21年の大日本教育会における「研究」の事業化過程
1898年大日本教育协会将“研究”商业化的过程
Effect of semen collection in extender solution on the characteristics of goat spermatozoa.
稀释液中精液采集对山羊精子特性的影响。
  • DOI:
  • 发表时间:
    2006
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Michihiro Hirayama;Yakov Pesin;Michihiro Hirayama;Michihiro Hirayama;白石崇人;白石崇人;白石崇人;白石崇人;白石崇人;白石崇人;Yamashiro H
  • 通讯作者:
    Yamashiro H
Transcripts encoding the enzymes that convert acetyl-CoA to cholesterol are induced in cumulus cells and are essential for progesterone biosynthesis and meiotic resumption of porcine oocvtes.
编码将乙酰辅酶A转化为胆固醇的酶的转录物在卵丘细胞中被诱导,并且对于猪卵母细胞的孕酮生物合成和减数分裂恢复至关重要。
  • DOI:
  • 发表时间:
    2005
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Michihiro Hirayama;Yakov Pesin;Michihiro Hirayama;Michihiro Hirayama;白石崇人;白石崇人;白石崇人;白石崇人;白石崇人;白石崇人;Yamashiro H;Yamashiro H;Yamashiro H;Wang HF;Yamashiro H;Shimada M
  • 通讯作者:
    Shimada M
Second Variational Formulae for Dimension Spectra
维数谱的二阶变分公式

Yakov Pesin的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Yakov Pesin', 18)}}的其他基金

Topics in Dynamical Systems: Attractors, Dimension, Lattice Models
动力系统主题:吸引子、维度、晶格模型
  • 批准号:
    1400027
  • 财政年份:
    2014
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Hyperbolic Dynamics, Large Deviations and Fluctuations
双曲动力学、大偏差和波动
  • 批准号:
    1300155
  • 财政年份:
    2013
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Topics in Dynamical Systems: Attractors, Dimension, Lattice Models
动力系统主题:吸引子、维度、晶格模型
  • 批准号:
    1101165
  • 财政年份:
    2011
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
TRAVEL SUPPORT FOR PARTICIPANTS OF PROGRESS IN DYNAMICS
为动态进步参与者提供旅行支持
  • 批准号:
    0968747
  • 财政年份:
    2009
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Topics in Dynamical Systems: Attractors, Dimension, Lattice Model
动力系统主题:吸引子、维度、晶格模型
  • 批准号:
    0754911
  • 财政年份:
    2008
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Workshop in Dynamical Systems and Related Topics
动力系统及相关主题研讨会
  • 批准号:
    0606947
  • 财政年份:
    2006
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Topics in Dynamical Systems: Attractors, Dimension, Lattice Model
动力系统主题:吸引子、维度、晶格模型
  • 批准号:
    0503810
  • 财政年份:
    2005
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Travel Support for Participants of International Conference "Kolmogorov and Contemporary Mathematics", June 16-21, 2003; Moscow, Russia
为2003年6月16-21日“柯尔莫哥洛夫与当代数学”国际会议与会者提供差旅费资助;
  • 批准号:
    0244702
  • 财政年份:
    2003
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Erodic Properties for 'Almost Hyperbolic' Systems
“几乎双曲”系统的侵蚀特性
  • 批准号:
    0196234
  • 财政年份:
    2001
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Topics in Dynamical Systems: Attractors, Dimension, Lattice Model
动力系统主题:吸引子、维度、晶格模型
  • 批准号:
    0088971
  • 财政年份:
    2000
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant

相似海外基金

Ergodic Properties of Smooth Systems on Manifolds
流形上光滑系统的遍历性质
  • 批准号:
    2247572
  • 财政年份:
    2023
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Applications of Descriptive Set Theory in Ergodic Theory and Smooth Dynamical Systems
描述集合论在遍历理论和光滑动力系统中的应用
  • 批准号:
    2100367
  • 财政年份:
    2021
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Ergodic Properties of Smooth Systems on Manifolds
流形上光滑系统的遍历性质
  • 批准号:
    1956310
  • 财政年份:
    2020
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Combinatorial constructions in Smooth Ergodic Theory
平滑遍历理论中的组合构造
  • 批准号:
    405305501
  • 财政年份:
    2018
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Research Fellowships
Ergodic Theory of Smooth One-Dimensional Maps
光滑一维映射的遍历理论
  • 批准号:
    1700291
  • 财政年份:
    2017
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
AMS Summer Research Institute on "Smooth Ergodic Theory and Applications
AMS 暑期研究所“平滑遍历理论与应用”
  • 批准号:
    9813732
  • 财政年份:
    1998
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Dynamical Systems and Smooth Ergodic Theory
动力系统和平滑遍历理论
  • 批准号:
    9803150
  • 财政年份:
    1998
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Dynamical Systems and Smooth Ergodic Theory
数学科学:动力系统和平滑遍历理论
  • 批准号:
    9204733
  • 财政年份:
    1992
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Topics in Dynamical Systems and Smooth Ergodic Theory
数学科学:动力系统和平滑遍历理论主题
  • 批准号:
    9011749
  • 财政年份:
    1990
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Dynamical Systems and Smooth Ergodic Theory
数学科学:动力系统和平滑遍历理论
  • 批准号:
    8905546
  • 财政年份:
    1989
  • 资助金额:
    $ 22.43万
  • 项目类别:
    Continuing Grant
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了