Canonical forms of shift-invariant maps on [N]∞[N]∞

We describe a canonical form for continuous functions Φ:[N]∞→[N]∞Φ:[N]∞→[N]∞ that commute with the shift map X↦X⧹{minX}. Then we investigate in which cases such a function ΦΦ satisfies that for every A∈[N]∞A∈[N]∞, there is X∈[N]∞X∈[N]∞ such that [X]∞⊆Φ″[A]∞[X]∞⊆Φ″[A]∞. This will lead us to solution of Problem 8.3 of [A.S. Kechris, S. Solecki, S. Todorcevic, Borel chromatic numbers, Adv. Math. 141 (1999) 1–44].