Study of ergodic properties of partially hyperbolic dynamical systems

部分双曲动力系统的遍历特性研究

• 批准号: 14340052
• 负责人: TSUJII Masato
• 金额: $ 4.22万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340052/
• 关键词: Dynamical system; Ergodic theory; Chaos; Partially hyperbolic dynamics; 力学系理論; 関数解析; 双曲力学系; 部分双曲的

In this study, we investigate ergodic properties of partially hyperbolic dynamicals systems. As one of the main results, we showed that generic partially hyperbolic endomorphism one two dimensional manifolds has finitely many physical measures (invariant measure that has attracting nature) such that the distribution of the orbits for almost every initial points with respect to the Lebesgue measure is asymptotic to one of those measures. Since this kind of results has only known for uniformly hyperbolic dynamical systems and one dimensional dynamical systems, this result is a kind of break-through. As a closely related subject, we also studied applications of functional analytic method in dynamical sytems. As a main result in this direction, we contructed a Banach spaces of distributions on which the Ruelle transfer operators acts on naturally and the essential spectral radius of that action is shown to be small. This lead to many results on decay of correlations and study of dynamical zeta functions.

本文研究了部分双曲动力系统的遍历性。作为主要结果之一，我们证明了二维流形上的一般部分双曲自同态具有许多物理测度（具有吸引性质的不变测度），使得几乎所有初始点的轨道关于Lebesgue测度的分布渐近于其中一个测度.由于这类结果只对一致双曲型动力系统和一维动力系统是已知的，因此这是一种突破。作为一个密切相关的课题，我们还研究了泛函分析方法在动力系统中的应用。作为这方面的一个主要结果，我们构造了一个Ruelle转移算子自然作用于其上的Banach分布空间，并且证明了该作用的本质谱半径是小的。这导致了许多结果衰减的相关性和动态zeta函数的研究。

项目成果

辻井正人: "Weakly expanding skew products of quadratic maps"Ergodic theory and dynamical systems. 23・5. 1401-1414 (2003)

Masato Tsujii：“二次映射的弱展开斜积” 遍历理论和动力系统 1401-1414（2003）。

Linearity of the exceptional set for maps of P_k(C)

P_k(C) 映射的异常集的线性

• 发表时间: 2004
• 期刊: Math. Ann. 330,no.1
• 作者: J.-Y. Briend;S. Cantat;M. Shishikura
• 通讯作者: M. Shishikura

Anisotropic Holder and Sobolev spaces for hyperbolic diffeom orphisms

双曲微分方程的各向异性 Holder 和 Sobolev 空间

• 发表时间: 2007
• 期刊: Annals of Institute Fourier 57
• 作者: J.Aaronson;II.Nakada;O.Sarig;辻井 正人
• 通讯作者: 辻井 正人

J.Buzzi: "Weakly expanding skew-products of quadratic maps"Ergodic theory and dynamical systems. (発表予定).

J.Buzzi：“二次映射的弱扩展斜积”遍历理论和动力系统（待介绍）。

Physical measures for partially hyperbolic surface endomorphisms

• DOI: 10.1007/bf02392516
• 发表时间: 2003-01
• 期刊: Acta Mathematica
• 影响因子: 3.7
• 作者: M. Tsujii