On finitely summable Fredholm modules from Smale spaces

We prove that allKK-homology classes of the stable (and unstable) Ruelle algebra a Smale space have explicit Fredholm module representatives are finitely summable on same smooth subalgebra and with degree summability. The is induced by metric underlying groupoid fine transversality relations between unstable sets. summability related to fractal dimension space. Further, modules obtained taking Kasparov products fundamental class Spanier-WhiteheadK-duality algebras. Finally, we obtain general results stability under holomorphic functional calculus construct Lipschitz algebras étale groupoids.