论文标题
田中 - ito $α$ - 连接的分数和匹配
Tanaka-Ito $α$-continued fractions and matching
论文作者
论文摘要
1981年，在1981年引入了两个密切相关的$α$连接部分的家庭：一方面是由田中和伊托（Ito）和伊托（Ito）。熵作为参数$α$的函数的行为已被广泛研究了中田的家族，并且已经获得了一些结果，利用了称为匹配的代数功能。在本文中，我们表明，匹配也发生在田中 - ito $α$ contin的分数，并且匹配间隔几乎完全涵盖了参数空间。实际上，匹配条件不存在的一组参数（称为分叉集）是零度量集（即使它具有完整的Hausdorff尺寸）。 Nakada的$α$连接的分数也共享了这一属性，但也存在一些实质性差异：田中 - ito持续分数的分叉设置不仅包含无限的许多理性值，还包含具有无界部分商的数字。
Two closely related families of $α$-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter $α$ has been studied extensively for Nakada's family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka-Ito $α$-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold, called bifurcation set, is a zero measure set (even if it has full Hausdorff dimension). This property is also shared by Nakada's $α$-continued fractions, and yet there also are some substantial differences: not only does the bifurcation set for Tanaka-Ito continued fractions contain infinitely many rational values, it also contains numbers with unbounded partial quotients.
