Universal minimal flows from pointwise completions and atomic decomposition

We organize explicit universal minimal flows of mapping and transformation groups around two standard reductions: passage to a compact group completion and removal of a closed extremely amenable normal subgroup. Finite interpolation identifies the pointwise completions of path, loop, gauge, and section groups; for disconnected compact Lie targets the completion also records the clopen information in the source. For measurable maps into a compact metrizable group K, the established atomless fixed-point theorem gives M(L0(Y, μ; K)) = KAμ, where Aμ is the set of atoms. Combining these reductions with known structure and compact-extension theorems yields atomic formulas for measure-space transformations, Lp-isometries, and automorphism groups of operator algebras. The general Banach-valued statement uses complex fibres; real Hilbert fibres are treated separately. We also give the formula M(FΩ ⋊ Sym(Ω)) = M(F)Ω × LO(Ω) for a Polish group F with metrizable universal minimal flow and a countable set Ω, with the natural skew action. Its ordered relational special cases have earlier Ramsey-theoretic treatments. The results are presented as explicit consequences and extensions of the cited completion, extreme-amenability, and structural results.

GitHub repository: https://github.com/eulogica/universal-minimal-flows