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Abstract: The mean ergodic theorem is equivalent to the assertion that for everyfunction K and every epsilon, there is an n with the property that the ergodicaverages A m f are stable to within epsilon on the interval n,Kn. We showthat even though it is not generally possible to compute a bound on the rate ofconvergence of a sequence of ergodic averages, one can give explicit bounds onn in terms of K and || f || - epsilon. This tells us how far one has to searchto find an n so that the ergodic averages are -locally stable- on a largeinterval. We use these bounds to obtain a similarly explicit version of thepointwise ergodic theorem, and show that our bounds are qualitatively differentfrom ones that can be obtained using upcrossing inequalities due to Bishop andIvanov. Finally, we explain how our positive results can be viewed as anapplication of a body of general proof-theoretic methods falling under theheading of -proof mining.-

Author: Jeremy Avigad, Philipp Gerhardy, Henry Towsner

Source: https://arxiv.org/

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