Complex and real topological dynamics

我们计划将Thurston的Mandelbrot集模型扩展到更高次的多项式上（例如，描述一个模型的边界的三次连通轨迹）和推广到更高的程度瑟斯顿的结果有关unlinked二次未成年人。我们想把一维谱分解推广到拓扑多项式上。此外，我们还研究了扩张多项式，包括c-帐篷平面映射（它将帐篷映射的联合收割机性质与复二次映射的性质结合起来，对于复二次映射应该起到与帐篷映射在区间上所起的作用相同的作用），并试图用基于映射的扩张性的工具来取代全纯工具。最后，我们计划描述所有点，使得它们周围的小扰动改变区间映射的动力学（这是稳定性的逐点版本），并进一步发展区间映射、复二次映射和台球的旋转理论。叠层的结果可能对其他领域产生强烈的影响，因为它们应该导致类似于Mandelbrot集的高维自相似集的模型;而且这一套是自然出现的。拓扑多项式的谱分解的模拟可以提供来自应用的复杂不可逆系统的极限行为的示例（例如，人口动态）。扩张多面体的结构结果将给出另一类映射，其类似于著名的复动力学结果。最后，在一维动力学的主题应该产生一个更好的理解一维映射的稳定性和方式不同类型的限制行为的一维映射共存。该项目可以产生教育影响，也可以对人力资源开发产生影响，因为其中的一些主题（如关于立方叠层、拓扑多项式的谱分解和扩展多项式的动力学的新成果/工具）可以作为研究生新课程的基础，并可以由学生在论文中开发（事实上，当地的一个研讨会专门讨论叠层）。

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).