## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 53, Number 3 (2012), 347-350.

### 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 $.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 53, Number 3 (2012), 347-350.

**Dates**

First available in Project Euclid: 25 September 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1348577704

**Digital Object Identifier**

doi:10.1215/00294527-1716757

**Mathematical Reviews number (MathSciNet)**

MR2981012

**Zentralblatt MATH identifier**

1258.03084

**Subjects**

Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]

Secondary: 37A25: Ergodicity, mixing, rates of mixing

**Keywords**

ergodic theorems computable analysis

#### Citation

Avigad, Jeremy. Uncomputably Noisy Ergodic Limits. Notre Dame J. Formal Logic 53 (2012), no. 3, 347--350. doi:10.1215/00294527-1716757. https://projecteuclid.org/euclid.ndjfl/1348577704