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 c2 σ-invariant maximal almost disjoint families on D. This implies that there are c2 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.