Operator Algebras for Multivariable Dynamics Kenneth

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 various topological conjugacy algebras; and two 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. The C*-envelope of A(X, τ) is the Cuntz-Pimsner algebra C * (X, τ) as defined by Katsura. We introduce a new concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise 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 isometrically) isomorphic if and only if the systems are piecewise topologically conjugate. We define a generalized notion of wandering sets and recurrence.