m at h . O A ] 7 M ay 2 00 7 OPERATOR ALGEBRAS FOR MULTIVARIABLE DYNAMICS

Let X be a locally compact Hausdorff space with n proper continuous self maps τ i : X → X for 1 ≤ i ≤ n. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra A(X, τ) and the semicrossed product C 0 (X) × τ F + n. We introduce a concept of conjugacy for multidimensional systems , which we coin piecewise conjugacy. We prove that the piece-wise conjugacy class of the system can be recovered from either the algebraic structure of A(X, τ) or C 0 (X) × τ F + n. Various classification results follow as a consequence. For example, for n = 2, 3, the tensor algebras are (algebraically or even completely isometri-cally) isomorphic if and only if the systems are piecewise topologi-cally conjugate. In order to establish these results we make use of analytic varieties as well as homotopy theory for Lie groups We define a generalized notion of wandering sets and recurrence.