Attractors, Smooth Dynamics and Combinatorics in Low-Dimension

摘要：Blokh计划描述某些有理函数的Milnor吸引子，确定它们的保形测度，并使用生长树的新概念作为研究Julia集合上动力学的工具（与Misiurewicz和Levin合作）。在一维动力学中，Blokh和Misiurewicz引入了超持久递归的概念。在某一点上缺乏这种递归是一种温和的膨胀性质。这种行为暗示了Blokh和Misiurewicz最近的结果，根据这些结果，具有负Schwarzian导数的野生吸引子是超持久循环的，并且在光滑区间映射的密集族中临界点的极限集是最小的。Blokh打算利用这个装置进一步研究光滑区间映射的吸引子性质和稳定性。他还希望将旋转理论扩展到图表上，然后研究周期轨道肖像中的强迫。生物学、物理科学和经济学中的许多过程都可以用动力系统来建模。研究这些模型可以让人们得出关于过程本身及其长期行为的有效结论。这个项目旨在研究各种这样的动力系统。Blokh打算验证系统的行为是平均混沌的（但在大多数初始条件下几乎是相同的），或者它与从一个特殊的有限的所谓临界初始条件开始的系统的行为一致。同样，如果相同的条件总是以相同的频率重复出现（例如，行星围绕太阳的运动），那么一个系统就被称为周期性行为。周期动力学比一般运动更容易研究，这就是为什么在探索更复杂系统的奥秘之前，尽可能全面地理解周期系统是很重要的。Blokh希望只用有限多个参数来描述动力系统的所有周期行为。所获得的结果将有助于描述各种过程的可能结果，并解释理论环境和自然世界中某些类型过程的普遍性。

Proposal: DMS-9970363Principal Investigator: Alexander M. BlokhAbstract: Blokh plans to describe Milnor attractors for certain rational functions, to determine their conformal measures, and to use the new notion of a growing tree as a tool for studying dynamics on the Julia set (jointly with Misiurewicz and Levin). In one-dimensional dynamics, Blokh and Misiurewicz introduced the notion of super-persistent recurrence. The lack of such recurrence at a point is a mild expanding property. This behavior implies recent results by Blokh and Misiurewicz according to which wild attractors with negative Schwarzian derivatives are super-persistently recurrent and limit sets of critical points in a dense family of smooth interval maps are minimal. Blokh intends to use this device to study further properties of attractors and stability for smooth interval maps. He also hopes to extend the rotation theory to graph maps and then study forcing among periodic orbit portraits.Many processes in biology, the physical sciences, and economics can be modeled by dynamical systems. Studying these models allows one to draw valid conclusions about the processes themselves and their long-term behavior. This project is aimed at studying a variety of such dynamical systems. Blokh intends to verify that either the behavior of the system is on average chaotic (but almost the same for the majority of initial conditions) or it coincides with the behavior of systems starting at a special finite set of so-called critical initial conditions. Also, a system is said to behave periodically if the same conditions recur with the same frequency all the time (e.g., the motion of the planets about the sun). Periodic dynamics are easier to study than more general motions, which is why it is important to understand periodic systems as fully as possible before probing the mysteries of more complicated systems. Blokh wishes to characterize all periodic behaviors of dynamical systems using only finitely many parameters. The results obtained will help to describe the possible outcomes for a variety of processes and to explain the prevalence of certain types of processes in both theoretical settings and the natural world.