  Let $X$ be compact Hausdorff, and $\phi: X \to X$ a continuous surjection.
Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the
relation fU = Uf\circ \phi$ or to the relation $Uf = f\circ \phi U.$ Then the
C$^*$-envelope of $\mathcal{A}$ is the crossed product of a commutative
C$^*$-algebra which contains $C(X)$ as a subalgebra, with respect to a
homeomorphism which we construct. We also show there are"sufficiently many"
nest representations.
