Primitive shifts on ψ-spaces

Let D be a countable discrete space and let p ∈ D . For any primitive shift σ : D → D ∖ { p } there are 2 c σ-invariant maximal almost disjoint families on D. This implies that there are 2 c pairwise non-homeomorphic Ψ ⁎ spaces admitting primitive shifts. Under a = c there exists a maximal almost disjoint family F on a countable discrete space D such that the spaces Ψ ( F ) and Ψ ⁎ ( F ) admit no primitive shift.