On the ergodic theory of Tanaka-Ito type alpha-continued fractions

Tanaka-Ito $α$-continued fractions and matching

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…

所有连分数之母的收缩诱导

Inducing contractions of the mother of all continued fractions

本文提出一类新的收缩Farey展开连分数算法，通过在Ito自然扩展中耦合诱导变换与收缩技术，统一了正则连分数、$S$-展开和Nakada的$α$-连分数，并给出其平面自然扩展的新描述。

正则连分数渐近分数与中间分数度量结果的一个统一理论

A unifying theory for metrical results on regular continued fraction convergents and mediants

本文通过重新审视 Ito 对 Farey 帐篷映射的自然扩张，建立了一个统一连分数度量理论的框架，将逼近系数极限分布、Legendre 型定理及 Lévy 定理推广等众多新旧结果统一为推论。