一类无限连分数变换的所有$α$ -变形的熵连续性
Continuity of entropy for all $α$-deformations of an infinite class of continued fraction transformations
摘要：
我们在Annali della Scuola Normale Pisa，Classe di Scienze上扩展了我们2020年论文的结果。在那里，我们关联到一个无限家庭的三角形富克斯群的一个单参数家庭的连分数地图，并表明，匹配（或同步）的间隔是充分的措施。在这里，我们找到每个映射的平面扩展，并证明了与每个单参数族相关联的熵函数的连续性。我们还引入了最终可扩区间映射的第一点态可扩幂的概念。我们证明了对于我们的单参数族中的每一个映射，它的第一个点态扩张幂映射的自然扩张由测地流到被相应群一致化的双曲轨道的单位切丛中的横截面的第一个返回给出。我们猜想这对我们所有的地图都成立。我们给出了数值证据的猜想。
英文摘要：
We extend the results of our 2020 paper in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. There, we associated to each of an infinite family of triangle Fuchsian groups a one-parameter family of continued fraction maps and showed that the matching (or, synchronization) intervals are of full measure. Here, we find planar extensions of each of the maps, and prove the continuity of the entropy function associated to each one-parameter family. We also introduce a notion of "first pointwise expansive power" of an eventually expansive interval map. We prove that for every map in one of our one-parameter families its first pointwise expansive power map has its natural extension given by the first return of the geodesic flow to a cross section in the unit tangent bundle of the hyperbolic orbifold uniformized by the corresponding group. We conjecture that this holds for all of our maps. We give numerical evidence for the conjecture.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
