Article

# Uncomputably noisy ergodic limits

Notre Dame Journal of Formal Logic (Impact Factor: 0.33). 05/2011; DOI: 10.1215/00294527-1716757

Source: arXiv

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**ABSTRACT:**Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p, where the constant C depends only on p and the modulus of uniform convexity. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. We clarify the relationship between bounds on the number of epsilon-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.Ergodic Theory and Dynamical Systems 03/2012; · 0.71 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We discuss the legacy of Alan Turing and his impact on computability and analysis.06/2012;

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