The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems

For typical properly ordered and minimal Bratteli diagrams $(B,{\le }_r)$, it is shown that there are finitely many invariant distributions ${\mathcal{D}}_i$ which are the only obstructions to solving the cohomological equation $f=u-u\circ \phi$ for the corresponding adic transformation $\phi :X_B\to X_B$ and for $\alpha$-Hölder $f$ with $\alpha$ large enough. These invariant distributions are then used to define cyclic cocycles, a.k.a. traces $\tau :K_0({\mathcal{A}}_{\phi })\to \mathbb{R}$ for the crossed product algebra ${\mathcal{A}}_{\phi }$.