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Martin-Löf Randomness, Invariant Measures and Countable Homogeneous Structures

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Abstract

We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fraïssé limits) to study closed amenable subgroups G of the symmetric group S ∞ of a countable set, where S ∞ has the topology of pointwise convergence. We construct G-invariant measures on the universal minimal flows associated with these groups G in, moreover, an algorithmic manner. This leads to an identification of the generic elements, in the sense of being Martin-Löf random, of these flows with respect to the constructed invariant measures. Along these lines we study the random elements of S ∞, which are permutations that transform recursively presented universal structures into such structures which are Martin-Löf random.

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Acknowledgements

The research is based upon work supported by the National Research Foundation (NRF) of South Africa. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and therefore the NRF does not accept any liability in regard thereto.

The author also wishes to express his thanks to an anonymous referee for a careful reading of this paper.

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  1. Department of Decision Sciences, University of South Africa, P.O. Box 392, 0003, Pretoria, South Africa

    Willem L. Fouché

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  1. Willem L. Fouché
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Fouché, W.L. Martin-Löf Randomness, Invariant Measures and Countable Homogeneous Structures. Theory Comput Syst 52, 65–79 (2013). https://doi.org/10.1007/s00224-012-9419-y

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