Algebraic characterization of multivariable dynamics

Abstract

Let X be a locally compact Hausdorff space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy.

In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.