Inverting the Furstenberg correspondence

Jeremy Avigad

doi:10.3934/dcds.2012.32.3421

Article Contents

2012, Volume 32, Issue 10: 3421-3431. Doi: 10.3934/dcds.2012.32.3421

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Inverting the Furstenberg correspondence

Jeremy Avigad^1,

1.
Departments of Philosophy and Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

Received: April 30, 2011

Revised: January 31, 2012

Published: September 2012

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Abstract

Given a sequence of sets $A_n \subseteq \{0,\ldots,n-1\}$, the Furstenberg correspondence principle provides a shift-invariant measure on $2^N$ that encodes combinatorial information about infinitely many of the $A_n$'s. Here it is shown that this process can be inverted, so that for any such measure, ergodic or not, there are finite sets whose combinatorial properties approximate it arbitarily well. The finite approximations are obtained from the measure by an explicit construction, with an explicit upper bound on how large $n$ has to be to yield a sufficiently good approximation.
We draw conclusions for computable measure theory, and show, in particular, that given any computable shift-invariant measure on $2^N$, there is a computable element of $2^N$ that is generic for the measure. We also consider a generalization of the correspondence principle to countable discrete amenable groups, and once again provide an effective inverse.

Keywords:

Mathematics Subject Classification: Primary: 37A45; Secondary: 03F60.

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