关于Minkowski测度
On the Minkowski Measure
摘要：
Minkowski问号函数将实数的连分数表示与它们的二进制展开联系起来。这个函数在很多方面都很特殊;一个是它的导数是“奇异的”。人们可以通过经典技巧证明，它的导数在所有有理数上必为零。由于问号本身是连续的，我们可以得出结论，导数在无理数上必须是非零的，因此是一个处处不连续的函数。这一衍生工具就是本文的主题。各种结果在这里：首先，一个简单的，但正式的测量理论建设的衍生物，使它清楚地表明，它有一个非常具体的存在作为一个勒贝格-斯蒂尔切斯测度，因此是安全的各种熟悉的方式进行操作。接下来，给出了一个精确的结果，将测度表示为分段连续函数的无穷乘积，每一段都是形式为（ax+b）/（cx+d）的莫比乌斯变换。这个结构然后被证明是一个特定的转移算子的Haar测度。给出了一个一般的证明，任何转移算子可以被理解为不超过也不小于一个推进的巴拿赫空间，这样的推进导致一个不变测度，哈尔测度，其中闵可夫斯基测度可以作为一个原型的例子。对它与Gauss-Kuzmin-Wirsing算子的关系作了一些小的说明。
英文摘要：
The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by classical techniques that its derivative must vanish on all rationals. Since the Question Mark itself is continuous, one concludes that the derivative must be non-zero on the irrationals, and is thus a discontinuous-everywhere function. This derivative is the subject of this essay. Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then shown to be the Haar measure of a certain transfer operator. A general proof is given that any transfer operator can be understood to be nothing more nor less than a push-forward on a Banach space; such push-forwards induce an invariant measure, the Haar measure, of which the Minkowski measure can serve as a prototypical example. Some minor notes pertaining to it's relation to the Gauss-Kuzmin-Wirsing operator are made.
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