On finitely summable Fredholm modules from Smale spaces

TL;DR

This paper constructs explicit finitely summable Fredholm modules representing all K-homology classes of Ruelle algebras associated with Smale spaces, linking summability to fractal dimensions and using Kasparov products.

Contribution

It provides explicit finitely summable Fredholm module representatives for K-homology classes of Ruelle algebras of Smale spaces, connecting summability to fractal geometry.

Findings

Fredholm modules are finitely summable on a smooth subalgebra

Summability degree relates to fractal dimension of the space

Constructs Lipschitz algebras on étale groupoids

Abstract

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