  A family of algebras, which we call topological conjugacy algebras, is
associated with each proper continuous map on a locally compact Hausdorff
space. Assume that $\eta_i:\X_i\to \X_i$ is a continuous proper map on a
locally compact Hausdorff space $\X_i$, for $i = 1,2$. We show that the
dynamical systems $(\X_1, \eta_1)$ and $(\X_2, \eta_2)$ are conjugate if and
only if some topological conjugacy algebra of $(\X_1, \eta_1)$ is isomorphic as
an algebra to some topological conjugacy algebra of $(\X_2, \eta_2)$. This
implies as a corollary the complete classification of the semicrossed products
$C_0(\X) \times_{\eta} \bbZ^{+}$, which was previously considered by Arveson
and Josephson, Peters, Hadwin and Hoover and Power. We also obtain a complete
classification of all semicrossed products of the form $A(\bbD)
\times_{\eta}\bbZ^{+}$, where $A(\bbD)$ denotes the disc algebra and $\eta:
\bbD \to \bbD$ a continuous map which is analytic on the interior. In this
case, a surprising dichotomy appears in the classification scheme, which
depends on the fixed point set of $\eta$. We also classify more general
semicrossed products of uniform algebras.
