Finitely Summable $K$-homology, the Index Pairing, and Cantor Minimal Systems

We study index pairings for crossed-product $C^{\ast }$-algebras arising from minimal actions on the Cantor set. We utilize Putnam’s orbit-breaking AF-subalgebras and embeddings to show we can compute any index pairing for Cantor minimal system crossed products using Connes’ trace formulas. In the case of odometers, we show that the associated algebras have uniformly finitely summable $K$-homology.