On spectral measures and convergence rates in von Neumann's Ergodic Theorem

M. Aloisio; S. L. Carvalho; C. R. de Oliveira; E. Souza

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arXiv · 2209.05290

On spectral measures and convergence rates in von Neumann's Ergodic Theorem

M. Aloisio·S. L. Carvalho·C. R. de Oliveira·E. Souza

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Abstract

We show that the power-law decay exponents in von Neumann's Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value~$1$. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann's Ergodic Theorem depend on sequences of time going to infinity.

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M. Aloisio, S. L. Carvalho, C. R. de Oliveira, E. Souza. 2022-09-12. On spectral measures and convergence rates in von Neumann's Ergodic Theorem. https://arxiv.org/abs/2209.05290

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