Dimension Groups for Polynomial Odometers

Here we study a class of dynamical systems we call polynomial odometers. These are adic maps on regularly structured Bratteli diagrams and include the Pascal and Stirling adic maps as examples. We describe the dimension groups associated with these systems and use this to study spaces of invariant measures. For many, but not all, the space of invariant measures is affinely homeomorphic to the space of Borel probability measures on a closed interval in $\mathbb{R}$ , we call such polynomial odometers reasonable. We describe the possible isomorphisms between dimension groups for reasonable polynomial odometers, and use this to prove a version of a result of Giordano, Putnam and Skau for this situation. Namely, we show that there is an isomorphism between unital ordered groups associated with two reasonable polynomial odometers if and only if there is a special kind of orbit equivalence between the two.