LOCAL STABILITY OF ERGODIC AVERAGES

By Jeremy Avigad, Philipp Gerhardy, Henry Towsner. 2007

103 citations recorded by the source corpus.

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue measure preserving transformation of [0,1] and a characteristic function f = XA such that the ergodic averages A n f do not converge to a computable element of L 2 ([0, 1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space and any element f, it is possible to compute a bound on the rate of convergence of 〈A n f〉 from T, f, and the norm ∥f*∥ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L 2 (X), then there is a computable bound on the rate of convergence of the sequence 〈Anf〉. The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every e > 0, there is an n with the property that the ergodic averages A m f are stable to within e on the interval [n, K(n)]. Even in situations where the sequence 〈A n f〉 does not have a computable limit, one can give explicit bounds on such n in terms of K and ∥f∥/e. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining.".

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8638561adbac68df66abd792922859de48ba3b67

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Citations in the corpus

Norm convergence of multiple ergodic averages for commuting transformations (2007, methodology): “ L2 metastability), and that the upper bound M∗ on Mis independent of P. The l= 1 version of this theorem was essentially established (with Zl P replaced by an arbitrary measure-preserving system) in [1]. CONVERGENCE OF MULTIPLE ERGODIC AVERAGES 3 Remark 1.8. The presence of the arbitrary function F”
Effective metastability of Halpern iterates in CAT(0) spaces (2011, background): “Based on the aforementioned logical metatheorems, [3] extracted the first explicit such uniform bound Φ for the mean ergodic theorem from its usual textbook proof.”
PROOF-THEORETIC METHODS IN NONLINEAR ANALYSIS (2019, methodology): “In [2], it is shown that in general there is no computable rate of convergence, but a primitive recursive rate of metastability is extracted using the proof-theoretic methods discussed above.”
Norm convergence of nilpotent ergodic averages (2011, result): “This type of statement already appears in the works of Tao [19] and of Avigad, Gerhardy and Towsner [1].”