四次多项式空间中的连通等熵猜想
The Connected Isentropes Conjecture in a Space of Quartic Polynomials
摘要：
这篇笔记是我2004年12月在石溪大学答辩的博士论文的缩略版。它说明了系统的动态复杂性如何在变形下演化。我考虑的对象是区间上的四次多项式映射，它们是两个逻辑斯蒂映射的合成。在这种映射的参数空间$P^{Q}$中，我考虑了与临界轨道是周期性的参数相对应的代数曲线，我把这种曲线称为左骨和右骨。使用准共形手术方法和刚度，我表明，骨头是简单的光滑弧连接两个边界点。我还详细分析了，使用揉捏理论，地图的组合学如何沿着骨骼演变。我们族中多项式的拓扑熵函数的行为与骨架的结构密切相关。本文的主要结论是所研究的参数空间中的熵水平集是连通的。
英文摘要：
This note is a shortened version of my dissertation thesis, defended at Stony Brook University in December 2004. It illustrates how dynamic complexity of a system evolves under deformations. The objects I considered are quartic polynomial maps of the interval that are compositions of two logistic maps. In the parameter space $P^{Q}$ of such maps, I considered the algebraic curves corresponding to the parameters for which critical orbits are periodic, and I called such curves left and right bones. Using quasiconformal surgery methods and rigidity I showed that the bones are simple smooth arcs that join two boundary points. I also analyzed in detail, using kneading theory, how the combinatorics of the maps evolves along the bones. The behavior of the topological entropy function of the polynomials in my family is closely related to the structure of the bone-skeleton. The main conclusion of the paper is that the entropy level-sets in the parameter space that was studied are connected.
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