随机性与非遍历系统
Randomness and Non-ergodic Systems
摘要：
我们的特点，满足Birkhoff的遍历定理在一定的可计算性条件下的算法随机性。首先，我们使用切割和堆叠的方法证明，如果康托空间的元素x不是马丁-洛夫随机的，则存在可计算的保测变换和可计算集合，证明x对于遍历定理来说不是典型的，这给了我们V 'yugin定理的逆。进一步证明了若x是弱2-随机的，则它对所有可计算的保测变换和所有下半可计算函数都满足遍历定理。
英文摘要：
We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.
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