Study on Complex dynamical systems and operator algebras

复杂动力系统与算子代数研究

• 批准号: 19340040
• 负责人: WATATANI Yasuo
• 金额: $ 7.07万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2010
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19340040/
• 关键词: 作用素環; 複素力学系; C*-環; 有理関数; 極大可換環; トレース; 作用素理; C^*-環; KMS状態

In the research we constructed C*-algebras of operators on Hilbert spaces for complex dynamical systems on a complex plane by the iteration of meromorphic functions like rational functions and transcendental entire functions. We found unexpected relations between complex dynamical systems and operator algebras. Self-similarity of Julia sets of complex dynamical systems corresponds to purely infinite simplicity of the associated C*-algebras. The branch structure of complex dynamical systems is related with the structure of KMS states (equilibrium states) for the gauge actions on the associated C*-algebras. We showed that we can recover the degree of the rational function, the number of branched points and the number of exceptional points.

本文通过有理函数和超越整函数等亚纯函数的迭代，构造了复平面上复动力系统的Hilbert空间上的算子C ~*-代数。我们发现了复杂动力系统和算子代数之间意想不到的关系。复杂动力系统的Julia集的自相似性对应于相关C*-代数的纯粹无限简单性。复杂动力学系统的分支结构与相应C ~*-代数上规范作用量的KMS态（平衡态）结构有关。我们证明了我们可以恢复有理函数的次数，分支点的数量和例外点的数量。

项目成果

Toeplitz-composition C^*-algebras for certain finite Blaschke products

某些有限 Blaschke 产品的 Toeplitz 组合 C^* 代数

• 发表时间: 2010
• 期刊: Proc.Amer.Math.Soc.
• 作者: H.Hamada;Y.Watatani
• 通讯作者: Y.Watatani

C^*-algebras associated with Mauldin-Williams graphs

与 Mauldin-Williams 图相关的 C^*-代数

• 发表时间: 2008
• 期刊: Canadian Mathematical Bulletin 51
• 作者: K.-I. Mitani;K.-S. Saito and T. Su zuki;M. Ionescu and Y. Watatani
• 通讯作者: M. Ionescu and Y. Watatani

Approximate innerness andCentral triviality of endomorphisms

近似内在性和自同态的中心琐碎性

• 发表时间: 2009
• 期刊: Adv.Math. 220
• 作者: T.Masuda;R.Tomatsu
• 通讯作者: R.Tomatsu

Composition operators and C*-algebras associated with complex dynamical systems

与复杂动力系统相关的复合算子和 C* 代数

• 发表时间: 2009
• 作者: H. Hamada;Y. Watatani;Y. Watatani
• 通讯作者: Y. Watatani

Spectral gap, and split property in quantum spin chains

量子自旋链中的光谱间隙和分裂特性

• 发表时间: 2010
• 期刊: J. Math. Phys
• 作者: Yin Yin Su Win;Y. Tsutsumi;T. Matsui
• 通讯作者: T. Matsui