NATURAL EXTENSION METHOD FOR TEH ERGODIC THOERY OF NUMBER THEORETIC TRANSFORMATIONS

1. 数论变换的正态数我们证明了正则连分数正态数的集合与最近的整数连分数正态数的集合是相同的。给出了b -正规数的定义，并证明了b -正规数的集合包含正规c - f正规数的集合。我们将这些结果的证明应用于F.Schweiger提出的正态数问题。我们对这个问题的回答是否定的。我们考虑了与Hecke群相关的后向连分数变换，并构造了它们的自然扩展。利用c.f.展开的周期性和有限性，给出了Hecke群的双曲点和椭圆点的表征。结果证明，这些是实数倒向连分式的自然推广。我们将数论变换的概念推广到非阿基米德域。首先我们考虑具有有限域系数的形式洛朗幂级数集合上的连分式。这里的连分式数字是具有有限域系数的多项式。接下来，我们开始研究f展开理论在这种情况下，我们将继续研究这个问题几年。该项目的主要部分将是研究非阿基米德领域的纤维系统理论。

1. Normal numbers of Number theoretic transformations We showed that the set of regular continued fraction normal numbers is identical with the set of the nearest integer continued fraction normal numbers. We also gave a definition of B-normal numbers and showed that the set of those numbers includes the set of regular c.f. normal numbers. We appliied the proof of these results to the normal number problem proposed by F.Schweiger. We gave a negative answer on this problem.2. We considered backward continued fractions transformations associated to Hecke groups and constructed their natural extensions. As a result, we gave characterizations of hyperbolic points and elliptic points of Hecke groups by the periodicity and the finiteness of the c.f. expansions. It turned out that these are natural generalizations of the backward continued fractions for real numbers.3. We extended the notion of number theoretic transformations to non-archimedian fields. First we consider continued fractions over the set of formal Laurent power seris with a finite field coefficients. Here the continued fraction digits are polynomials with the finite field coefficients. Next, we started to study f-expansion theory in this case and we will continue for some years on this problem. Main part of the project will be to study the theory of Fibered Systems fornon- archimedian fileds.