ON INFINITE MEASURE PRESERVING MEASURABLE DYNAMICAL SYSTEMS

本课题的主要结果有三个：一个保单位测度变换作为一个具有有限测度集的导出变换，可以化为保有限测度变换。在这种情况下，ceiling函数表示返回到这个集合的时间，并且它是不可积的。在这个意义上，我们研究了具有无穷期望的随机过程。特别是，我们研究了连续分数混合随机过程几乎无限的期望。在随机变量分布的一定条件下，证明了随机变量分布的轻修剪后的强大数定律，并研究了遍历非奇异变换的Maharum扩张。我们讨论了与无理旋转和子移位相关的Maharam扩张的局部有限不变测度。在有限马尔可夫子移位的情况下，我们用子移位的共形测度刻画了这种测度。作为无穷遍历理论的应用，我们研究了算术级数理论中的Erods猜想。在这个观点下，我们考虑了无穷保测度变换的多重递归性质。我们给出了角谷型多重数的较低估计。我们还研究了度量数论作为遍历理论的应用。主要结果如下：（1）证明了Jacobi-Perron算法收敛性的算术分布的一个性质。(2)我们构造了与罗森连分式相关的Farey映射.这里的Farey映射是1维映射，它诱导连分数的中位数收敛。

The following three are the main results of this project:An unite measure preserving transformation can be reduced to a finite measure preserving one as an induced transformation with a set of finite measure. In this case, the ceiling function represents the return time to this set and it is non-integrable. In this sense, we studied the stochastic processes with infinite expectations. In particular, we studied continued fraction mixing stochastic processes with barely infinite expectations. We have proved the strong law of large numbers after light trimming under some conditions on the distribution of random variables.We also studied Maharum extension of ergodic nonsingular transformations. We discussed locally finite invariant measures for Maharam extensions associated to irrational rotations and subshifts. In the case of finite Markov subshifts, we characterized such measures by conformal measures for subshifts. We extend such results to countable Markov subshifts.As an application of infinite ergodic theory, we studied the theory of arithmetic progressions in particular, concerning to Erods conjecture. In this point of view, we considered the multiple recurrence property of infinite measure preserving transformations. We gave an lower estimate of the multiplicity by Kakutani type. We also studied metric number theory as an application of ergodic theory. Some of main results are the following (1) We proved a property of the arithmetic distribution of convergents arising from the Jacobi-Perron algorithm. (2) We constructed Farey maps associated to Rosen's continued fractions. here the Farey maps are 1-dimensional maps which induces mediant convergents of continued fractions.