(This is joint work with Elias Katsoulis.) Let X be a compact Hausdorff space with n continuous maps of X into itself. To this we associate various topological conjugacy algebras; and two emerge as the natural candidates for the universal algebra of the system. The classification of these algebras up to isomorphism leads to a new notion which we call piecewise conjugacy. Generalized notions of wandering sets and recursion are used to characterize when these algebras are semisimple.