Dynamical systems of type (m, n) and their C*-algebras

Given positive integers n and m, we consider dynamical systems in which (the disjoint union of) n copies of a topological space is homeomorphic to m copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra denoted by O-m,O-n, which in turn is obtained as a quotient of the well-known Leavitt C*-algebra L-m,L-n, a process meant to transform the generating set of partial isometries of L-m,L-n into a tame set. Describing O-m,O-n as the crossed product of the universal (m, n)-dynamical system by a partial action of the free group Fm+n, we show that O-m,O-n is not exact when n and m are both greater than or equal to 2, but the corresponding reduced crossed product, denoted by O-m,n(r), is shown to be exact and non-nuclear. Still under the assumption that m, n >= 2, we prove that the partial action of Fm+n is topologically free and that O-m,n(r) satisfies property (SP) (small projections). We also show that O-m,n(r) admits no finite-dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.