ON THE INVARIANT MEASURE FOR THE TRANSFORMATIONS ASSOCIATED WITH SOME REAL CONTINUED-FRACTIONS.
ON THE INVARIANT MEASURE FOR THE TRANSFORMATIONS ASSOCIATED WITH SOME REAL CONTINUED-FRACTIONS.
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关于与某些实连分数相关的变换的不变测度。
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发表时间：
1977
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通讯作者：
Shigeru Tanaka
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作者：
H. Nakada;Shunji Ito;Shigeru Tanaka
我们引入了两种类型的真实的连分式展开式，其中之一是HURWITZ的复连分式展开式的真实的部分。对于与这些展开式有关的变换，我们将根据P. Levy对简单连分式的情形的方法确定不变测度的精确形式。此外，我们将阐明P. Levy方法的数学意义。§ 0引言在研究空间上给定的可测变换的性质时，变换下的测度不变（如果存在）提供了一个有价值的线索。因此，在这样的调查中，人们经常采取以下方法。首先，人们问变换是否有一个不变的措施拥有合理的性质。其次，如果存在这样一个不变测度，人们就试图确定它具体形式。当然，一般来说，很难得到一个不变测度的精确形式，因为人们必须从有关的每个变换的精确描述中预测地得到它。另一方面，正是由于这个原因，如果能够推导出不变测度的具体形式，那么它对于给定变换的定量分析是极其有用的。159 H1 Tosrn NAKADA，S11 uNJI 1 To和SmGERU TANAKA在这方面，我们回顾了与著名的简单连分式展开引起的变换有关的非凡历史。对于这种变换，高斯指出，具有1 - 2 - 1形式的密度函数的测度是不变的。实际上，og +x，如果给定函数--1-._ _1 __，则很容易证明它是一个测度不变量在简单连分式变换下的log 2l+：c密度。然而，历史捉弄了我们，让我们没有线索，至于高斯实际上是如何达到这个功能的-1 - 1-。很久以后，KuzMIN [3]和LEVY [4] log 2 l+x在他们各自的论文中给出了获得不变测度的密度函数1 1 2 1 1的方法，并填补了这一空白，尽管我们无法知道GAt~ss所使用的推理是否与KL：ZMIN和LEVY所使用的推理相同。在本文中，我们制定，然后解决了几个问题。第一个问题是寻找精确确定简单连分式变换及相关变换的不变测度的有效方法。第二个问题是要澄清数学结构背后的方法所使用的利维在他的推导密度函数1 - 1 - 2 1 1 -。og +x考虑到这些目标，本文的结构如下：在§ 1中，我们简化了Lf：vy在[ 4]中给出的论证，导出了与简单连分式展开有关的变换的不变测度的具体形式。然而，我们在这一节中使用的方法是基于一个相当技术性和似乎限制性的假设，我们将在这一点上不予解释。因此，我们将§ 1中采用的方法称为“基于纯粹偶然发现的方法”。在§ 2中，我们引入了一种新型的（真实的）连分数展开。这种展开对应于HuRWITZ在[2]中引入的复连分数展开的真实的部分。在§ 3中，我们引入了另一种真实的连分式展开。这两个连分式展开式之间的关系可以用以下方式来解释：每个连分式展开式都以自然的方式在正整数的无限序列空间（符号空间）上导出一个自同态。其中一个自同态的自然扩张是另一个自同态的自然扩张的逆。由于这些原因，我们将把§ 3中定义的变换称为与§ 2中定义的变换相关联的向后变换。在§ 4中，我们将确定§ 2和§ 3中定义的连分式变换的不变测度的密度函数的精确形式。我们在§ 4中的方法不同于§ 1中所采用的“纯机会发现方法”。事实上，我们希望我们在第4节中的程序将澄清数学意义，并为看似特别的“纯偶然发现方法”提供正当性。为了证明这一主张，我们将在§ 5中表明，函数g（x）似乎突然出现并以某种不自然的方式出现的原因是因为对于简单连分式展开的情况，由它引起的变换的不变测度上的变换160和它的向后变换彼此重合，正是这一事实使得很难解释函数g（x）出现的自然性。在结束这些介绍性发言时，我们要感谢TAKUJI 0 NOYAMA教授，Yu 11 ho教授和YmcHmo高桥教授对这个问题的关注和宝贵的建议。§ 1连分式变换与高斯的不变测度众所周知，与简单连分式展开式相关的变换T定义如下：对于xE[O，1），
We introduce two types of real continued-fraction expansions, one of which is the real part of the complex continued-fraction expansion of HURWITZ. For the transformations associated to these expansions we shall determine the precise form of invariant measures according to the method of P. LEVY for the case of simple continued-fraction. Moreover, we shall clarify the mathematical meaning of the method of P. LEVY. § 0 Introduction In the investigation of properties of a measurable transformation given on a space, a measure invariant under the transformation, if it exists, provides a valuable clue. Hence, one often takes the following approach in such an investigation. First, one asks whether the transformation has an invariant measure possessing reasonable properties. Next, if there is sucn an invariant measure, one tries to determine its concrete form. Of course, it is, in general, difficult to obtain the precise form of an invariant measure since one has to obtain it predictly from the precise description of each transformation concerned. On the other hand, for this very reason, the derivation of the concrete form of an invariant measure, if it can be carried out, is extremely useful for the quantitative analysis of the given transformation. 159 H1Tosrn NAKADA, S11uNJI lTo and SmGERU TANAKA In this connection, we recall that there is remarkable history associated with the transformation induced by the well-known simple continued-fraction expansion. For this transformation GAuss pointed out as if it is obvious apriori that the measure having the density function of the form 1 -} 2 --1 lis invariant. Indeed, og +x if one is given the function --1-.___ _1 __ , then it is easy to prove that it is the log 2 l+:c density of a measure invariant under the simple continued-fraction transformation. However, history played a trick and left us with no clue as to how GAuss actually arrived at this function --1----1-. Much later, KuzMIN [3] and LEVY [4] log 2 l+x showed, in their respective papers, ways to arrive at the density function 1 1 2 1 1 for the invariant measure and filled this missing gap, although we og +x have no way of knowing whether the reasoning used by GAt~ss was the same as those employed by KL:ZMIN and LEVY. In this paper, we formulate and then solve a couple of problems. The first problem is to search for effective methods for determining precisely the invariant measure for simple continued-fraction transformation and other related transformations. The second problem is to clarify the mathematical structure lying behind the method used by LEVY in his derivation of the density function 1 -1-2 1 1 -. og +x With these objectives in mind, we structure this paper in the following manner: In § 1 we simplify Lf:vy's argument give in [ 4] to derive the concrete form of the invariant measure for the transformation associated with simple continued-fraction expansion. The method we employ in this section, however, is based on a rather technical and seemingly restrictive assumption, which we shall leave unexplained at that point. For this reason, we shall call the method employed in § 1 "the method based on pure chance discovery ". In § 2 we introduce a new type of (real) continued-fraction expansion. This expansion corresponds to the real part of the complex continued-fraction expansion introduced by HuRWITZ in [2]. In § 3, we introduce still another type of real continued-fraction expansion. The relationship between these two continued-fraction expansions can be explained in the following way: Each continued-fraction expansion induces in a natural way an endomorphism on the space of infinite sequences of positive integer (symbolic space). The natural extension of one of these endomorphisms turns out to be the inverse of the natural extension of the other one. For these reason, we shall call the transformation defined in § 3 the backward transformation associated with transformation defined in § 2. In § 4, we will determine the precise form of the density function for the invariant measures for the continued-fraction transformations defined in § 2 and § 3. Our method in § 4 is different from the "method of pure-chance discovery" employed in § 1. We hope, in fact, that our procedure in § 4 will clarify the mathematical meaning and give justification to seemingly ad-hoc "method of pure chance discovery ". In order to justify this claim, we shall show in § 5 that the reason why the function g(x) seems to emerge suddenly and in somewhat unnatural manner is because for the case of simple continued-fraction expansion the transformation 160 On the Invariant Measure for the Transformations Associated induced by it and its backward transformation coincide with each other, and that it is this fact which makes it difficult to explain the naturalness of the emergence of the function g(x). In concluding these introductory remarks, we would like to thank Professors TAKUJI 0NOYAMA, Yu11 ho and YmcHmo TAKAHASHI for their interest on the problem and valuable advice. § 1 The Transformation Associated with Continued-Fraction and the Invariant Measure of GAuss As it is well-known, the transformation T associated with simple continuedfraction expansions is defined as follows : For xE[O, 1),