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A constructive view on ergodic theorems

Published online by Cambridge University Press:  12 March 2014

Bas Spitters
Bas Spitters*
Affiliation:
Radboud University, Nijmegen, The Netherlands. E-mail: B.Spitters@cs.ru.nl
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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
Information
The Journal of Symbolic Logic , Volume 71 , Issue 2 , June 2006 , pp. 611 - 623
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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