Dimension Groups and Invariant Measures for Polynomial Odometers

Adic maps on simple ordered Bratteli diagrams, called Bratteli-Vershik systems, have been proven to be extremely useful as models for minimal Cantor systems. In particular, the dimension group given by the diagram is order isomorphic to the ordered cohomology group for the system, and this along with a distinguished order unit is shown to be the complete invariant for strong orbit equivalence in [8], see also [9, 14]. A wider class of Bratteli-Vershik systems involves adic maps on potentially non-simple ordered Bratteli diagrams. These examples include the well-studied Pascal adic system [13, 16, 17, 20], the Stirling adic system [7, 10], and the Euler adic system [2, 6, 10, 18]. Although these are examples of adic maps on non-simple, non-stationary Bratteli diagrams, the diagrams are highly structured, and thus these seem natural extensions of the minimal Cantor systems to study. In addition, there are connections between these systems and reinforced random walks [7, 10]. In this paper we unify the study of some of these examples by examining a class which we call polynomial odometers. These are adic maps defined by a sequence of polynomials with positive integer coefficients. This class includes the Pascal and Stirling (but not the Euler) adic systems as examples as well as the classical odometers. As we have defined them here, the adic maps of this type are defined everywhere, but there are countably many points of discontinuity unless the system is a classical odometer. Except for the classical odometer, these systems are not minimal. In the non-minimal case but there is a rich supply of fully supported invariant measures. In the next two sections of the paper we establish basic definitions and facts about polynomial odometers. In particular, we show in Theorem 5 that for a polynomial odometer (X,T ), the dimension group of the diagram is order isomorphic to C(X,Z)/(∂TC(X,Z) ∩ C(X,Z)) where ∂TC(X,Z) = {g ◦ T − g | g ∈ C(X,Z)}.