Oscillation and the mean ergodic theorem

Details Oscillation and the mean ergodic theorem Oscillation and the mean ergodic theorem

2012-03-19

arxiv.org/abs/1203.4124v1

Mathematics

Dynamical Systems

Scientific paper

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 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.

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