一维动力系统中若干问题的研究

项目名称： 一维动力系统中若干问题的研究

项目编号： No.11271344

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李思敏

作者单位： 中国科学技术大学

项目金额： 68万元

中文摘要： 一维动力系统理论是动力系统理论的一个重要组成部分，除了自身丰富的研究内涵外，也可为高维系统提供方法和思路。本项目将从组合、拓扑、测度等多个角度出发，结合复动力系统中的思想和方法，研究区间映射的若干动力系统问题，包括：利用principal nest刻画映射临界点轨道的组合性质；研究wild attractor的随机稳定性、存在性及Hausdorff维数的下界估计；考察绝对连续不变测度的等价刻画命题的优化及向多临界点映射的推广；研究区间映射Julia集的双曲维数与最小共形指数的关系等。这些问题与申请人前期的工作有着密切的联系，是前期工作进一步的延续和深化。本项目的创新之处在于：首次提出用principal nest刻画映射的组合型；丰富了研究多临界点区间映射的方法与技巧；注重复动力系统在区间映射动力系统中的应用。项目的实施将丰富和完善一维动力系统的研究成果，为将来进一步的研究提供借鉴和思路。

中文关键词： 单峰映射；Cantor 吸引子；绝对连续不变测度；随机稳定；复杂性函数

英文摘要： The theory of one dimensional dynamical system is an important component of the theory of dynamical systems. Besides the fruitful research content in this area, it also provides method and idea for high dimensional dynamics. This project will study some problems in one dimensional dynamics from combinatorical, topological and measure theoritical viewpoint, combined with the idea and method from complex dynamical systems. It includes: Describe the combinatorial properties of critical orbits by principal nest; Study the stochastic stability，existence and estimation of Hausdorff dimension from below of the wild attractor; Optimize the theorem on the equivalent description of the existence of the absolutely continuous invariant measure and improve it to multi-modal maps; Study the relationships between the hyperbolic dimension of Julia set and the minimal exponent of the conformal measure. These problems have close connection with the applicant's previous works and can be seen as the continuation of previous works. The innovation of this project is: the idea of describing the combinatorial property by principal nest is new; it enriches the method and technique for dealing with multi-modal maps; it emphasise the application of complex dynamics in interval dyanmics. These research will enrich and improve the known res

英文关键词： unimodal maps；Cantor attractor；absolutely continuous invariant measure；random stable；complexity function