This is an announcement for the paper "Oscillation and the mean ergodic theorem" by Jeremy Avigad and Jason Rute.
Abstract: Let B be a uniformly convex Banach space, let T be a nonexpansive linear operator, and let A_n x denote the ergodic average (1/n) sum_{i<n} T^n x. A generalization of the mean ergodic theorem due to Garrett Birkhoff asserts that the sequence (A_n x) converges, which is equivalent to saying that for every epsilon > 0, the sequence has only finitely many fluctuations greater than epsilon. Drawing on calculations by Kohlenbach and Leustean, we provide a uniform bound on the number of fluctuations that depends only on rho := || x || / epsilon and a modulus, eta, of uniform convexity for B. Specifically, we show that the sequence of averages (A_n x) has O(rho^2 log rho * eta(1/(8 rho))^{-1})-many epsilon-fluctuations, and if B is a Hilbert space, the sequence has O(rho^3 log rho)-many epsilon-fluctuations. The proof is fully explicit, providing a remarkably uniform, quantitative, and constructive formulation of the mean ergodic theorem.
Archive classification: math.DS math.FA math.LO
Mathematics Subject Classification: 37A30, 03F60
Submitted from: avigad@cmu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1203.4124
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