The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems

TL;DR

This paper investigates the cohomological equation for minimal Cantor systems derived from Bratteli diagrams, identifying invariant distributions that serve as obstructions and using them to define cyclic cocycles in the associated algebra.

Contribution

It establishes the finiteness of invariant distributions obstructing solutions to the cohomological equation and constructs cyclic cocycles from these distributions.

Findings

Finitely many invariant distributions obstruct the cohomological equation.

Invariant distributions are used to define cyclic cocycles for the algebra.

Results apply to typical properly ordered minimal Bratteli diagrams.

Abstract

For typical properly ordered and minimal Bratteli diagrams $(B,\leq_r)$, 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 $\phi:X_B\rightarrow X_B$ and for $\alpha$-H\"older $f$ with $\alpha$ large enough. These invariant distributions are then used to define cyclic cocycles, a.k.a. traces $\tau:K_0(\mathcal{A}_\phi)\rightarrow \mathbb{R}$ for the crossed product algebra $\mathcal{A}_\phi$.