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

TL;DR

This paper links the decay rates in von Neumann's Ergodic Theorem to spectral measure exponents and explores how convergence rates depend on time sequences in systems without a spectral gap.

Contribution

It establishes a connection between decay exponents and spectral measures and analyzes convergence rates without spectral gaps under weak convergence assumptions.

Findings

Decay exponents are spectral measure pointwise scaling exponents.

Convergence rates depend on sequences of time in systems without spectral gaps.

Weak convergence assumptions influence the ergodic convergence behavior.

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.