LOCAL STABILITY OF ERGODIC AVERAGES
LOCAL STABILITY OF ERGODIC AVERAGES
复制标题
遍历平均值的局部稳定性
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
H. Towsner
中科院分区：
文献类型：
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作者：
J. Avigad;P. Gerhardy;H. Towsner
我们考虑在何种程度上可以计算的遍历平均序列的收敛速度的界限。不难构造一个[0，1]的可计算勒贝格测度保持变换和一个特征函数f = XA的例子，使得遍历平均值Anf不收敛于L2（[0，1]）的可计算元素.特别是，对于该序列的收敛速度没有可计算的界限。另一方面，我们证明了，对于可分Hilbert空间上的任何非扩张线性算子T和任何元素f，都可以从T，f和极限的范数f * f* f计算出收敛速度的界。特别地，如果T是由概率空间X的可计算遍历测度保持变换产生的Koopman算子，并且f是L2（X）的任何可计算元素，则存在序列Δ Anf Δ的收敛速度的可计算界。平均遍历定理等价于这样的断言：对于每个函数K（n）和每个e > 0，存在具有遍历平均值A m f在区间[n，K（n）]上的e内稳定的性质的n。即使在序列A n f没有可计算极限的情况下，我们也可以根据K和f/e给出这样的n的明确界限。这告诉我们要搜索多远才能找到一个n，使得遍历平均在一个大的区间上是“局部稳定的”。我们使用这些界限来获得逐点遍历定理的类似明确版本，并且我们表明我们的界限与使用Bishop和Ivanov的上交不等式获得的界限在性质上不同。最后，我们解释了我们的积极成果可以被视为一个机构的一般证明理论的方法属于“证明挖掘”的标题下的应用。".
We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0,1] and a characteristic function f = XA such that the ergodic averages A n f do not converge to a computable element of L 2 ([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space and any element f, it is possible to compute a bound on the rate of convergence of 〈A n f〉 from T, f, and the norm ∥f*∥ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L 2 (X), then there is a computable bound on the rate of convergence of the sequence 〈Anf〉. The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every e > 0, there is an n with the property that the ergodic averages A m f are stable to within e on the interval [n, K(n)]. Even in situations where the sequence 〈A n f〉 does not have a computable limit, one can give explicit bounds on such n in terms of K and ∥f∥/e. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining.".