Complex and real topological dynamics

复杂而真实的拓扑动力学

• 批准号: 1201450
• 负责人: Alexander Blokh
• 金额: $ 13.5万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201450&HistoricalAwards=false
• 关键词: Complex; real; topological; dynamics

We plan to extend Thurston's model of the Mandelbrot set onto polynomials of higher degrees (e.g., describe a model of the boundary of the cubic connectedness locus) and generalize onto higher degrees Thurston's results about unlinked quadratic minors. We want to extend the one-dimensional spectraldecomposition for topological polynomials. In addition, we study expanding polymodials, including c-tent plane maps (which combine properties of tent maps with properties of complex quadratic maps and should play the same role for quadratic complex maps as tent maps play on the interval) and seek to replace holomorphic tools by tools based upon the expansiveness of maps. Finally, we plan to describe all points such that small perturbations around them change the dynamics of the interval map (this is a pointwise version of stability) and to further develop rotation theory for interval maps, complex quadratic maps, and billiards.The results on laminations may have a strong impact on other areas as they should lead to a model of a higher dimensional self-similar set analogous to the Mandelbrot set; moreover, this set appears in a natural fashion. The analog of spectral decomposition for topological polynomials may provide examples of limit behavior for complicated non-invertible systems from applications (e.g., from population dynamics). Structural results on expanding polymodials would give another class of maps for which analogs of well-known complex dynamical results hold. Finally, topics in one-dimensional dynamics should yield a better understanding of stability of one-dimensional maps and the way different types of limit behavior of a one-dimensional map coexist. The project can have an educational impact as well as impact upon human resources development as some of the topics (such as new results/tools on cubic laminations, on spectral decomposition for topological polynomials and on dynamics of expanding polymodials) could serve as the basis of new courses for graduate students and could be developed by students in their dissertations (in fact, a local seminar is devoted to laminations).

我们计划将瑟斯顿的Mandelbrot集模型推广到高次多项式(例如，描述三次连通性轨迹的边界的模型)，并将瑟斯顿关于不相连的二次子式的结果推广到更高阶。我们想推广拓扑多项式的一维谱分解。此外，我们还研究了扩展多面体，包括C-帐篷平面映射(它结合了帐篷映射的性质和复二次映射的性质，对于二次复映射应该起到与帐篷映射在区间上的作用相同的作用)，并寻求用基于地图的可扩充性的工具来取代全纯工具。最后，我们计划描述所有的点，使它们周围的微小扰动改变区间映射的动力学(这是逐点稳定性的版本)，并进一步发展区间映射、复二次映射和台球的旋转理论。关于层叠的结果可能会对其他领域产生强烈的影响，因为它们应该导致一个类似于Mandelbrot集的高维自相似集的模型；而且，这个集以一种自然的方式出现。拓扑多项式的谱分解的模拟可以从应用(例如，从种群动力学)提供复杂不可逆系统的极限行为的例子。关于扩张多面体的结构结果将给出另一类映射，这类映射类似于众所周知的复杂动力学结果。最后，一维动力学的主题应该更好地理解一维映射的稳定性以及一维映射的不同类型的极限行为共存的方式。该项目可以对人力资源开发产生教育影响和影响，因为一些主题(例如关于三次分层、拓扑多项式的谱分解和扩展多面体的动力学的新结果/工具)可以作为研究生新课程的基础，并且可以由学生在他们的论文中开发(事实上，当地专门举办了关于分层的研讨会)。