慢连分数、变换器和Serret定理
Slow continued fractions, transducers, and the Serret theorem
摘要：
连分数初等理论中的一个基本结果是：两个实数的连分数展开式具有相同的尾部当且仅当它们在PGL（2，Z）的射影作用下属于同一轨道.这个结果最早是在Serret's Cours d 'algèbre supériminals中提出的，所以我们将把它称为Serret定理。尽管在文献中有丰富的连分数算法，但似乎缺少对Serret结果的统一处理。本文证明了在一个大的算法族中，由Gauss映射的分支生成的PGL（2，Z）的子群Sigma有1000多种可能性，并且证明了实数的每个Sigma-等价类可划分为1000多种尾等价类，并给出了尾等价类的个数的界.我们的方法是通过有限状态传感器，高斯映射相互关联。它们构成了群的Schreier图的opfibrations，它们的同步性--可能成立也可能不成立--保证了a.e. Serret定理的有效性
英文摘要：
A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). This result was first formulated in Serret's Cours d'algèbre supérieure, so we'll refer to it as to the Serret theorem. Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. In this paper we show that there are finitely many possibilities for the subgroups Sigma of PGL(2,Z) generated by the branches of the Gauss maps in a large family of algorithms, and that each Sigma-equivalence class of reals is partitioned in finitely many tail-equivalence classes, whose number we bound. Our approach is through the finite-state transducers that relate Gauss maps to each other. They constitute opfibrations of the Schreier graphs of the groups, and their synchronizability ---which may or may not hold--- assures the a.e. validity of the Serret theorem.
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