V’yugin has shown that there are a computable shift-invariant measure on and a simple function such that there is no computable bound on the rate of convergence of the ergodic averages . 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 .
"Uncomputably Noisy Ergodic Limits." Notre Dame J. Formal Logic 53 (3) 347 - 350, 2012. https://doi.org/10.1215/00294527-1716757