同步是一类无限连分数变换的所有$α$ -变形的完全测度
Synchronization is full measure for all $α$-deformations of an infinite class of continued fraction transformations
摘要：
本文研究了一类与三角形富克斯群相关的区间映射的单参数变形的无限族，即所谓的$α$ -连分数。通常，对于这种单参数变形，给出由$α$索引的映射的熵的函数以与映射的端点的轨道是否同步直接相关的方式变化。对于与模群$\text{PSL}_2(\mathbb Z)$的经典情形相关联的单参数变形的两种情况，已经确定了同步发生的$α$的集合。在这里，我们明确地确定同步集的每个$α$ -变形在我们的无限家庭。(In一般来说，我们的富克斯群不是模群的子群，因此将$α$ -展开与正则连分数展开联系起来的工具对我们来说是不可用的。）这里的一个奇怪之处是，我们所有的同步集都可以用一个单词树来描述。在准备的一篇论文中，我们确定了我们的地图的自然扩张，以及与每个变形相关的熵函数。
英文摘要：
We study an infinite family of one-parameter deformations, so-called $α$-continued fractions, of interval maps associated to distinct triangle Fuchsian groups. In general for such one-parameter deformations, the function giving the entropy of the map indexed by $α$ varies in a way directly related to whether or not the orbits of the endpoints of the map synchronize. For two cases of one-parameter deformations associated to the classical case of the modular group $\text{PSL}_2(\mathbb Z)$, the set of $α$ for which synchronization occurs has been determined. Here, we explicitly determine the synchronization sets for each $α$-deformation in our infinite family. (In general, our Fuchsian groups are not subgroups of the modular group, and hence the tool of relating $α$-expansions back to regular continued fraction expansions is not available to us.) A curiosity here is that all of our synchronization sets can be described in terms of a single tree of words. In a paper in preparation, we identify the natural extensions of our maps, as well as the entropy functions associated to each deformation.
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