The C$^*$-envelope of a semicrossed product and nest representations

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 $fU = Uf\circ \phi$. 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.