Transformations and Colorings of Groups

Abstract

AbstractLet G be a compact topological group and let f : G → G be a continuous transformation of G. Define f* : G → G by f*(x) = f (x–1)x and let μ = μG be Haar measure on G. Assume that H = Im f* is a subgroup of G and for every measurable C ⊆ H, μG(( f*)–1(C)) = μH(C). Then for every measurable C ⊆ G, there exist S ⊆ C and g ∈ G such that f (Sg–1) ⊆ Cg–1 and μ(S) ≥ (μ(C))2.