论文标题
瑟斯顿定理：维度一个
Thurston's Theorem: Entropy in Dimension One
论文作者
论文摘要
在他的论文中，瑟斯顿（Thurston）表明，当且仅当其扩展常数$λ= e^h $的情况下，代表自由组的外部自动形态的巨像的火车Track的拓扑熵是一个弱的perron数字。这是一个有力的结果，回答了一个类似于伪anosov同态的表面和拉伸因素的问题。但是，用于证明这一在Traintrack地图上进行开创性定理的许多机器都包含在瑟斯顿论文的一部分中，内容涉及后有限的间隔图的熵，并且很难解析证据。在这份说明性论文中，我们将瑟斯顿的方法现代化，填补原始论文中的空白，然后提炼瑟斯顿的方法，以提供Traintrack定理的凝聚力证明。特别值得注意的是增加了Traintrack代表的崇高证明，这在Thurston的论文中缺少。
In his paper, Thurston shows that a positive real number $h$ is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant $λ= e^h$ is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.
