Let A(Δ) be the disc algebra. A
Dirichlet subalgebra of A(Δ) is obtained by forming A(p) = A(Δ) ∩{f ∘p : f ∈ A(Δ)}
where p(eit) in a singular homeomorphism of the unit circle. In this paper
the endomorphisms of A(p) are described and for a restricted class of these
endomorphisms the spectra is computed.