Operator algebras for multivariable dynamics

Abstract

Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $τ_i:X \to X$ for $1 \le i \le n$. 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 $\rC_0(X)\times_τ\Fn$. We introduce a 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 $\rC_0(X)\times_τ\Fn$. 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. 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. Using this, it is shown that $\A(X, τ)$ or $\rC_0(X)\times_τ\Fn$ is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each $τ_i$ being surjective and $v$-recurrent points being dense for each $v \in \Fn$.