Borel chromatic numbers of graphs of commuting functions

Let D = (X,D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0, . . . , Fn−1 : X → X are Borel functions, let DF0,...,Fn−1 be the directed graph that they generate. It is an open problem if χB(DF0,...,Fn−1) ∈ {1, . . . , 2n + 1,א0}. This was verified for commuting functions with no fixed points. We show here that for commuting functions with the properties that χB(DF0,...,Fn−1) < א0 and that there is a path from each x ∈ X to a fixed point of some Fj , there exists an increasing filtration {Xm}m<ω with X = ⋃ m<ωXm such that χB(DF0,...,Fn−1 Xm) ≤ 2n for each m. We also prove that if n = 2 in the previous case, then χB(DF0,F1) ≤ 4. It follows that the approximate measure chromatic number χ ap M (D) does not exceed 2n+ 1 when the functions commute.