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

关于II型∞和III型遍历变换的澄清问题及其应用

基本信息

批准号：
10440060
负责人：
NAKADA Hitoshi
金额：
$ 6.14万
依托单位：
KEIO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B).
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-10440060/
关键词：
NONSINGULAR TRANSFORMATION MARKOV SHIFT MULTIPLE RECURRENCE FORMAL POWER SERIES II_∞型エルゴード変換 dissipative sequence upper Banach density weakly wandering sequence Zの直和分解 Pascal-adic変換 2次元ランドム・ウォーク Rogers-Ramanujan連分数展開 II_∞型変換 III_

项目摘要

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.

1. II 型 ∞ 遍历变换的递归重数：我们通过返回序列和 Kakutani-Parry 指数对马尔可夫移位集进行分类。这个结果出现在以色列期刊 J.Aaronson 和 H.Nakada 的论文中。 Math. 2000。此外，Hamachi 的研究小组已经证明，紧群扩展可以保留递归的重数。 2．如果基本变换的旋转数是有界类型，则圆柱流的局部有限遍历不变量测度的集合是连续的，这些测度是从圆的 PL 同胚中导出的，在该项目中，这些测度是由 Herman 考虑的。我们证明了这种圆柱流没有其他局部有限不变测度。另一方面，我们考虑了马尔可夫型加法机的 Maharam 扩展。我们还确定了局部有限遍历集。与Hoelder连续势相关的Maharam扩展的不变测度。3．我们继续对有限域系数的形式幂级数展开。此外，我们还考虑了丢潘汀近似的度量理论，并证明了一些经典度量定理的类比。特别是，我们证明了 Dufine-Schaeffer 定理的形式幂级数版本。