Classifying a Family of Models for the Henon Attractor: A U.S. - Croatia Collaboration

动力系统拓扑学研究的主要内容之一是研究动力系统(如吸引子)下左不变集的拓扑结构。我们建议研究非双曲H‘{e}非吸引子的一族模型的拓扑结构：由区间的单峰映射生成的逆极限空间。逆极限拓扑论中的一个著名猜想是Ingram猜想，它指出捏合序列是由单峰映射生成的逆极限的拓扑不变量。我们的目的是对由单峰映射生成的一大类逆极限进行分类，即由具有稀疏后临界轨道的帐篷映射生成的逆极限空间。这将是证明英格拉姆猜想的重要一步。在过去的几十年里，对混沌动力系统的研究已经变得越来越重要。研究的主要途径之一是研究混沌吸引子的微妙结构。这一努力的一个主要障碍是，通常我们没有对这些吸引子的精确数学描述，而是我们只有数字证据。然而，对于某些混沌吸引子，有一类模型确实有一个精确的数学描述：单峰映射的逆极限空间。更充分地了解这些逆极限空间的结构将有助于更好地理解混沌吸引子。这项提议代表了与克罗地亚萨格勒布大学的国际合作。合作的目的是完整地描述某些混沌吸引子的这一族模型的结构。

One of the main thrusts of research in the topology of dynamical systems is an investigation into the topological structure of sets left invariant under a dynamical system, such as attractors. We propose investigating the topological structure of a family of models for the non-hyperbolic H\'{e}non attractors: inverse limit spaces generated by unimodal maps of the interval. A celebrated conjecture in the topological theory of inverse limits is the Ingram Conjecture which states that the kneading sequence is a topological invariant for inverse limits generated by unimodal maps. Our aim is to classify a large family of inverse limits generated by unimodal maps, namely the inverse limit spaces generated by tent maps with a sparse postcritical orbit. This will be a major step towards a proof of the Ingram Conjecture.The study of chaotic dynamical systems has surged in importance in the last few decades. One of the main avenues of research is into the delicate structure of chaotic attractors. A major stumbling block in this endeavor is that usually we do not have a precise mathematical description for these attractors, but rather we have only numerical evidence. However, there is a family of models for certain chaotic attractors that do have a precise mathematical description: inverse limit spaces of unimodal maps. Understanding the structure of these inverse limit spaces more fully will lead to a better understanding of chaotic attractors. This proposal represents an international collaboration with the University of Zagreb, Croatia. The aim of the collaboration is to fully describe the structure of this family of models for certain chaotic attractors.