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

动态系统拓扑研究的重点之一是研究动态系统下左不变集合的拓扑结构，如吸引子。我们提出研究非双曲H\'{e}非吸引子的一类模型的拓扑结构：由区间的单峰映射生成的逆极限空间。逆极限拓扑理论中一个著名的猜想是Ingram猜想，该猜想指出，对于由单峰映射生成的逆极限，揉捏序列是一个拓扑不变量。我们的目的是对单峰映射生成的一大群逆极限进行分类，即具有稀疏临界后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，混沌动力系统的研究变得越来越重要。研究混沌吸引子的精细结构是研究混沌吸引子的主要途径之一。这一努力的一个主要障碍是，通常我们对这些吸引子没有精确的数学描述，而是只有数值证据。然而，对于某些混沌吸引子，有一组模型确实具有精确的数学描述：单峰映射的逆极限空间。更充分地理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。这项建议是与克罗地亚萨格勒布大学的国际合作。合作的目的是充分描述这类模型对某些混沌吸引子的结构。

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.