Uncomputably Noisy Ergodic Limits

Abstract

V’yugin has shown that there are a computable shift-invariant measure on $2^{\mathbb{N}}$ 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 $\epsilon$.