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von neumann’s ergodic theorem

Ergodic Theory and Dynamical Systems (1)

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Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap
Part of
JONATHAN BEN-ARTZI, BAPTISTE MORISSE
Journal:

Ergodic Theory and Dynamical Systems / Volume 41 / Issue 6 / June 2021

Published online by Cambridge University Press:

13 April 2020, pp. 1601-1611

Print publication:

June 2021
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Von Neumann’s original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time averages of solutions are shown.