Martin-Lof randomness, invariant measures and countable homogeneous structurs

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.