Article

# Uncomputably Noisy Ergodic Limits

(Impact Factor: 0.36). 05/2011; 53(3). DOI: 10.1215/00294527-1716757
Source: arXiv

ABSTRACT

V'yugin has shown that there are a computable shift-invariant measure on
Cantor space and a simple function f such that there is no computable bound on
the rate of convergence of the ergodic averages A_n f. Here it is shown that in
fact one can construct an example with the property that there is no computable
bound on the complexity of the limit; that is, there is no computable bound on
how complex a simple function needs to be to approximate the limit to within a
given epsilon.

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• "Corollary 6.3. There is a separable, reflexive, and strictly convex Banach space B, such that for every u, there is an x ∈ B with x ≤ 1 such that the sequence (A n x) has (1/4)-fluctuations in each of the intervals [1] [2], [2] [4], . . . , [2 u−1 , 2 u ]. "
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