Martin-Lof randomness, invariant measures and countable homogeneous structurs

TL;DR

This paper explores the connection between Martin-Löf randomness, invariant measures, and countable homogeneous structures by constructing invariant measures on universal minimal flows of certain groups and analyzing their random elements.

Contribution

It introduces an algorithmic method to construct invariant measures on universal minimal flows of closed amenable subgroups of the symmetric group and identifies generic elements as Martin-Löf random.

Findings

Invariant measures can be constructed algorithmically for these flows.

Generic elements in the flows are characterized as Martin-Löf random permutations.

Random permutations transform universal structures into Martin-Löf random structures.

Abstract

We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fra\" {i}ss\' e limits) to study closed amenable subgroups $G$ of the symmetric group $S_\infty$ of a countable set, where $S_\infty$ 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-L\" of random, of these flows with respect to the constructed invariant measures. Along these lines we study the random elements of $S_\infty$, which are permutations that transform recursively presented universal structures into such structures which are Martin-L\" of random.