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arXiv · 2410.08124

The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems

Rodrigo Treviño
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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 $\mathcal{A}_\phi$.

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Rodrigo Treviño. 2024-10-10. The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems. https://arxiv.org/abs/2410.08124

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