A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.

中文翻译：

---

区间映射的混沌边界

研究区间上的动力学的目标是了解向正拓扑熵的过渡。从1980年代开始有一个推测，达到正拓扑熵的唯一途径是通过级联倍增的分叉级联。我们在光滑间隔图的自然族中证明了这个猜想，并用它来研究具有正熵的映射边界的结构。特别是，我们显示出在具有固定数量的关键点的映射族中，边界是本地连接的，对于解析映射，它是一个单元集。