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Two quantitative notions of mixing are the decay of correlations and thedecay of a mix-norm -- a negative Sobolev norm -- and the intensity of mixingcan be measured by the rates of decay of these quantities. From duality,correlations are uniformly dominated by a mix-norm; but can they decayasymptotically faster than the mix-norm? We answer this question byconstructing an observable with correlation that comes arbitrarily close toachieving the decay rate of the mix-norm. Therefore the mix-norm is thesharpest rate of decay of correlations in both the uniform sense and theasymptotic sense. Moreover, there exists an observable with correlation thatdecays at the same rate as the mix-norm if and only if the rate of decay of themix-norm is achieved by its projection onto low-frequency Fourier modes. Inthis case, the function being mixed is called q-recurrent; otherwise it isq-transient. We use this classification to study several examples and raisequestions for future investigations.

On mix-norms and the rate of decay of correlations