On a clacification problem of type II ∞ and type III ergodic transformations and its application

1.第II类∞遍历变换的重复性：我们根据返回序列和Kakutani-Parry指数对马尔可夫移位集进行分类。这一结果发表在J.Aaronson和H.Nakada在《以色列数学杂志》上的一篇论文中。2000年。此外，Hamachi的研究小组已经证明，通过紧群扩张，常返性的重数是保持的。已知圆柱流的局部有限遍历不变测度集的基数是连续的，如果基变换的旋转数是有界型的。这些度量是由Herman所考虑的圆的PL同胚导出的。在这个项目中，我们证明了对于这样的圆柱流，不存在其他的局部有限不变测度。另一方面，我们考虑了马尔可夫型加法机的Maharam扩张。我们还确定了这类与霍尔德连续势相关的Maharam扩张的局部有限遍历不变测度集。研究了有限域系数形式幂级数的连分式展开式。此外，我们还考虑了折射近似的度规理论具有正特征。我们证明了一些经典度量定理的类比成立。特别地，我们证明了DuFine-Schaeffer定理的形式幂级数形式。

1. Multiplicity of recurrence of type II ∞ ergodic transformations : We classified the set of Markov shifts by the return sequence and also by Kakutani-Parry index. This result was appeared in a paper by J.Aaronson and H.Nakada, Israel Journal of Math. 2000. Moreover, Hamachi's research group has shown that the multiplicity of recurrence is preserved by the compact group extensions.2. It has been known that the cardinality of the set of locally finite ergodic nvariant measures for a cylinder flow is continuous if the rotaion number of the base transformation is of bounded type. These measure are induced from the PL homeomorphisms of the circle, those were considered by Herman. In this project, we proved that there is no other locally finite invariant measure for such cylinder flows. On the other hand, we considered Maharam extensions of adding machines of Markovian type. We also determined the set of locally finite ergodic invariant measures for such Maharam extensions associated to Hoelder continuous potentials.3. We studied continued fraction expansions of formal power series with a finite field coefficients. Moreover we considered the metrical theory of diopantine approximation in positive characteristic. We showed that analogue of some classical metric theorems hold. In particular, we proved the formal power series version of Dufine-Schaeffer thoerem.