Attractors, Smooth Dynamics and Combinatorics in Low-Dimension

提议：DMS-9970363首席研究员：Alexander M.Blokh摘要：Blokh计划描述某些有理函数的Milnor吸引子，确定它们的共形度量，并将正在生长的树的新概念用作研究Julia集上的动力学的工具(与Misiurewicz和Levin合作)。在一维动力学中，Blokh和Misiurewicz引入了超持久重现的概念。在某一点上不再出现这种情况，是一种温和的扩张性质。这一行为暗示了Blokh和Misiurewicz最近的结果，即具有负Schwarzian导数的野生吸引子是超持久循环的，并且稠密光滑区间映射族中的临界点的极限集是极小的。Blokh打算利用这个工具进一步研究吸引子的性质和光滑区间映射的稳定性。他还希望将旋转理论扩展到图形地图，然后研究周期轨道之间的作用力。生物学、物理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对过程本身及其长期行为得出有效的结论。这个项目的目的是研究各种这样的动力系统。Blokh打算验证系统的行为要么是平均混沌的(但对于大多数初始条件几乎是相同的)，要么是与从所谓的关键初始条件的特殊有限集合开始的系统的行为一致。此外，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)，那么系统就被认为是周期性的。周期动力学比更一般的运动更容易研究，这就是为什么在探索更复杂系统的奥秘之前，尽可能全面地了解周期系统是很重要的。Blokh希望仅使用有限多个参数来刻画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，并解释某些类型的过程在理论环境和自然界中的盛行。

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.