Uncomputably noisy ergodic limits

TL;DR

This paper demonstrates that for certain ergodic limits, not only is the convergence rate uncomputable, but the complexity of approximating the limit also cannot be bounded by any computable function, highlighting fundamental limits in ergodic theory.

Contribution

It constructs an example showing that the complexity of ergodic limits can be uncomputably unbounded, extending previous results on convergence rates.

Findings

No computable bound on the convergence rate of ergodic averages.

No computable bound on the complexity of the limit function.

Highlights fundamental limits in the computability of ergodic limits.

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.