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Uniform convergence in von Neumann’s ergodic theorem in the absence of a spectral gap

Part of: Ergodic theory Hyperbolic equations and systems Equations of mathematical physics and other areas of application Smooth dynamical systems: general theory

Published online by Cambridge University Press:  13 April 2020

JONATHAN BEN-ARTZI Open the ORCID record for JONATHAN BEN-ARTZI [Opens in a new window]  and
BAPTISTE MORISSE
JONATHAN BEN-ARTZI
Affiliation:
School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK email [email protected], [email protected]
BAPTISTE MORISSE
Affiliation:
School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK email [email protected], [email protected]
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Abstract

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.

Keywords

Von Neumann’s ergodic theoremconvergence ratesdensity of states

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 37A25: Ergodicity, mixing, rates of mixing 37A10: One-parameter continuous families of measure-preserving transformations 37C40: Smooth ergodic theory, invariant measures 35Q41: Time-dependent Schrödinger equations, Dirac equations 35L05: Wave equation
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1601 - 1611
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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