Journal of Symbolic Logic

A constructive view on ergodic theorems

Bas Spitters

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Let T be a positive L₁-L contraction. We prove that the following statements are equivalent in constructive mathematics.

  • 1. The projection in L₂ on the space of invariant functions exists;
  • 2. The sequence (Tⁿ)n ∈ N Cesáro-converges in the L₂ norm;
  • 3. The sequence (Tⁿ)n ∈ N Cesáro-converges almost everywhere.
Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.

As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop.

Article information

J. Symbolic Logic Volume 71, Issue 2 (2006), 611-623.

First available in Project Euclid: 2 May 2006

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Spitters, Bas. A constructive view on ergodic theorems. J. Symbolic Logic 71 (2006), no. 2, 611--623. doi:10.2178/jsl/1146620162.

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  • Jeremy Avigad and Ksenija Simic, Fundamental notions of analysis in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, to appear.
  • Errett Bishop, Mathematics as a numerical language, Intuitionism and Proof Theory (Proceedings of the summer conference at Buffalo, N.Y., 1968), North-Holland, Amsterdam,1970, pp. 53--71.
  • Errett Bishop and Douglas Bridges, Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer-Verlag,1985.
  • Errett A. Bishop, Foundations of constructive analysis, McGraw-Hill Publishing Company, Ltd.,1967.
  • N. Dunford and J. T. Schwartz, Linear operators. Part I: General theory, Interscience Publishers,1958.
  • Hajime Ishihara and Luminiţa Vîţǎ, Locating subsets of a normed space, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 10, pp. 3231--3239.
  • Ulrich Krengel, Ergodic theorems, Studies in Mathematics, de Gruyter,1985.
  • J. A. Nuber, A constructive ergodic theorem, Transactions of the American Mathematical Society, vol. 164 (1972), pp. 115--137.
  • --------, Erratum to `A constructive ergodic theorem', Transactions of the American Mathematical Society, vol. 216 (1976), p. 393.
  • Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press,1983.
  • Bas Spitters, Constructive and intuitionistic integration theory and functional analysis, Ph.D. thesis, University of Nijmegen,2002.
  • Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag,1982.