On the small boundary property and Z-absorption, II

Consider a minimal and free topological dynamical system (X, Zᵈ). It is shown that zero mean dimension of (X, Zᵈ) is characterized by Z-absorption of the crossed product C*-algebra A=C(X) Zᵈ, where Z is the Jiang-Su algebra. In fact, among other conditions, the following are shown to be equivalent: (1) (X, Zᵈ) has the small boundary property. (2) A A ⊗ Z. (3) A has uniform property Γ. (4) l^∞(A)/J2, ω, T(A) has real rank zero. The same statement also holds for unital simple AH algebras with diagonal maps.