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A constructive view on ergodic theorems

Published online by Cambridge University Press:  12 March 2014

Bas Spitters*
Affiliation:
Radboud University, Nijmegen, The Netherlands. E-mail: B.Spitters@cs.ru.nl

Abstract

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Avigad, Jeremy and Simic, Ksenija, Fundamental notions of analysis in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, to appear.Google Scholar
[2]Bishop, Errett, Mathematics as a numerical language, Intuitionism and Proof Theory (Proceedings of the summer conference at Buffalo, N.Y., 1968). North-Holland, Amsterdam, 1970. pp. 5371.CrossRefGoogle Scholar
[3]Bishop, Errett and Bridges, Douglas, Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer-Verlag, 1985.CrossRefGoogle Scholar
[4]Bishop, Errett A., Foundations of constructive analysis, McGraw-Hill Publishing Company, Ltd., 1967.Google Scholar
[5]Dunford, N. and Schwartz, J. T., Linear operators. Part I: General theory, Interscience Publishers, 1958.Google Scholar
[6]Ishihara, Hajime and Vîǎţ, Luminita, Locating subsets of a normed space, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 10, pp. 32313239.CrossRefGoogle Scholar
[7]Krengel, Ulrich, Ergodic theorems, Studies in Mathematics, de Gruyter, 1985.CrossRefGoogle Scholar
[8]Nuber, J. A., A constructive ergodic theorem, Transactions of the American Mathematical Society, vol. 164 (1972), pp. 115137.CrossRefGoogle Scholar
[9]Nuber, J. A., Erratum to ‘A constructive ergodic theorem’. Transactions of the American Mathematical Society, vol. 216 (1976), p. 393.Google Scholar
[10]Petersen, Karl, Ergodic theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1983.CrossRefGoogle Scholar
[11]Spitters, Bas, Constructive and intuitionistic integration theory and functional analysis, Ph.D. thesis. University of Nijmegen, 2002.Google Scholar
[12]Walters, Peter, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, 1982.CrossRefGoogle Scholar