Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-16T11:14:57.762Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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 Ben-ArtziJ@cardiff.ac.uk, MorisseB@cardiff.ac.uk
BAPTISTE MORISSE
Affiliation:
School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK email Ben-ArtziJ@cardiff.ac.uk, MorisseB@cardiff.ac.uk
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Assani, I. and Lin, M.. On the one-sided ergodic Hilbert transform. Contemp. Math. 430 (2007), 21–39.CrossRefGoogle Scholar
Ben-Artzi, M. and Klainerman, S.. Decay and regularity for the Schrödinger equation. J. Anal. Math. 58(1) (1992), 25–37.CrossRefGoogle Scholar
Cycon, H. L., Froese, R. G., Kirsch, W. and Simon, B.. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Springer, Berlin, 1987.CrossRefGoogle Scholar
Dzhulaii, N. A. and Kachurovskii, A. G.. Constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem with continuous time. Sib. Math. J. 52(5) (2011), 824–835.CrossRefGoogle Scholar
Evans, L. C.. Partial Differential Equations (Graduate Studies in Mathematics, 19) . American Mathematical Society, Providence, RI, 2010.Google Scholar
Kachurovskii, A. G.. The rate of convergence in ergodic theorems. Russian Math. Surveys 51(4) (1996), 653–703.CrossRefGoogle Scholar
Kachurovskii, A. G.. On uniform convergence in the ergodic theorem. J. Math. Sci. 95(5) (1999), 2546–2551.Google Scholar
Kachurovskii, A. G.. Fejér integrals and the von Neumann ergodic theorem with continuous time. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 474 (2018), 171–182.Google Scholar
Kachurovskii, A. G. and Knizhov, K. I.. Deviations of Fejer sums and rates of convergence in the von Neumann ergodic theorem. Dokl. Math. 97(3) (2018), 211–214.CrossRefGoogle Scholar
Kachurovskii, A. G. and Podvigin, I. V.. Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems. Trans. Moscow Math. Soc. 77 (2016), 1–53.CrossRefGoogle Scholar
Kachurovskii, A. G. and Podvigin, I. V.. Fejér sums for periodic measures and the von Neumann ergodic theorem. Dokl. Math. 98(1) (2018), 344–347.CrossRefGoogle Scholar
Kachurovskii, A. G. and Reshetenko, A. V.. On the rate of convergence in von Neumann’s ergodic theorem with continuous time. Sb. Math. 201(4) (2010), 493–500.CrossRefGoogle Scholar
Kachurovskii, A. G. and Sedalishchev, V. V.. On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem. Math. Notes 87(5–6) (2010), 720–727.CrossRefGoogle Scholar
Kachurovskii, A. G. and Sedalishchev, V. V.. Constants in estimates for the rates of convergence in von Neumann’s and Birkhoff’s ergodic theorems. Sb. Math. 202(8) (2011), 1105–1125.CrossRefGoogle Scholar
Kato, T.. Perturbation Theory for Linear Operators. Springer, Heidelberg, 1995.CrossRefGoogle Scholar
von Neumann, J.. Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. USA 18(2) (1932), 70–82.CrossRefGoogle Scholar
Riesz, F.. Some mean ergodic theorems. J. Lond. Math. Soc. (2) s1‐13(4) (1938), 274–278.CrossRefGoogle Scholar
Tao, T.. Nonlinear Dispersive Equations: Local and Global Analysis (Regional Conference Series in Mathematics, 106) . American Mathematical Society/CBMS, Providence, RI, 2006.CrossRefGoogle Scholar
Tataru, D.. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients, II. Amer. J. Math. 123(3) (2001), 385–423.CrossRefGoogle Scholar