轨道理论、局部有限置换和Morse算法
Orbit theory, locally finite permutations and morse arithmetic
摘要：
本文从遍历轨道理论的角度分析了两个起源于组合数论的保测变换。我们研究Morse变换（在整数并元数的群$\textbf{Z}_2$中的进实现，如作者[{ \sl J. Sov.数学。{ \textbf{28} }，667-674（1985）; { \sl St. Petersburg Math. J.} { \textbf{6} }（1995），no. 3，529-540]），并证明它具有与并矢里程计相同的轨道划分。然后，我们给出了一个精确的描述的时间替代的里程表，它产生的莫尔斯变换。以群$\mathbb Z$的随机重新排序的形式，或以群$\mathbb Z$的随机无限排列的形式来描述这种时间置换是方便的。我们引入了{ \it局部有限置换（LFP）或局部有限双射（LFB），以及对$\mathbb Z$群（以及所有顺从群）的一致局部有限时间置换（ULFTS）}的概念.如果两个具有相同轨道划分的自同构的时间替换是ULFTS，则称它们为{ \it allied}。我们的主要结果是莫尔斯变换和里程表是结盟的。群$\mathbb Z$（以及更一般的群）上的随机无限置换理论与群作用的遍历论一样强。这一领域的主要任务是研究无穷置换，无穷置换空间上的测度，以及$\mathbb Z$上的线性序。局部有限置换类是这种分析的一个有用的类。
英文摘要：
The goal of this paper is to analyze two measure preserving transformation of combinatorial and number-theoretical origin from the point of view of ergodic orbit theory. We study the Morse transformation (in its adic realization in the group $\textbf{Z}_2$ of integer dyadic numbers, as described by the author [{\sl J. Sov. Math.} {\textbf{28}}, 667-674 (1985); {\sl St. Petersburg Math. J.} {\textbf{6}} (1995), no. 3, 529-540]) and prove that it has the same orbit partition as the dyadic odometer. Then we give a precise description of time substitution of the odometer, which produces the Morse transformation. It is convenient to describe this time substitution in the form of random re-orderings of the group $\mathbb Z$, or in terms of random infinite permutations of the group $\mathbb Z$. We introduce the notion of {\it locally finite permutations (LFP) or locally finite bijection (LFB), and uniformly locally finite time substitution (ULFTS)} for the group $\mathbb Z$ (and for all amenable groups). Two automorphisms which have the same orbit partitions are called {\it allied} if the time substitution of one to another is ULFTS. Our main result is that the Morse transformation and the odometer are allied. The theory of random infinite permutations on the group $\mathbb Z$ (and on more general groups) is as strong as the ergodic theory of actions of the group. The main task in this area is the investigation of infinite permutations, and measures on the space of infinite permutations, as well as the study of linear orderings on $\mathbb Z$. The class of locally finite permutations is a useful class for such an analysis.
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