NATURAL EXTENSION METHOD FOR TEH ERGODIC THOERY OF NUMBER THEORETIC TRANSFORMATIONS

点击翻译按钮获取中文摘要

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.