On mix-norms and the rate of decay of correlations
On mix-norms and the rate of decay of correlations
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DOI：
10.1088/1361-6544/abdbbd
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发表时间：
2021-06-01
期刊：
影响因子：
1.7
通讯作者：
Doering, Charles R.
中科院分区：
文献类型：
--
作者：
Oakley, Bryan W.;Thiffeault, Jean-Luc;Doering, Charles R.
混合的两个定量概念是相关性的衰减和混合范数的衰减--负的索伯列夫范数--混合的强度可以通过这些量的衰减率来测量。从对偶性来看，相关性一致地被一个混合范数所支配;但是它们能比混合范数渐近地衰减得更快吗？我们通过构造一个具有相关性的可观测量来回答这个问题，该可观测量任意接近于实现混合范数的衰减率。因此，混合范数是一致意义和渐近意义下相关性衰减的最快速率。此外，存在一个具有相关性的可观测量，它以与混合范数相同的速率衰减，当且仅当混合范数的衰减速率是通过它在低频傅里叶模上的投影实现的。在这种情况下，被混合的函数被称为q -递归函数;否则它是q-瞬态函数。我们使用这种分类来研究几个例子，并提出问题，为今后的调查。
Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm-a negative Sobolev norm-and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q -recurrent; otherwise it is q-transient. We use this classification to study several examples and raise questions for future investigations.