# On finitely summable Fredholm modules from Smale spaces

@inproceedings{Gerontogiannis2021OnFS, title={On finitely summable Fredholm modules from Smale spaces}, author={D. M. Gerontogiannis}, year={2021} }

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…

## References

SHOWING 1-10 OF 102 REFERENCES

Compact metric spaces, Fredholm modules, and hyperfiniteness

- MathematicsErgodic Theory and Dynamical Systems
- 1989

Abstract We show that the existence of a finitely summable unbounded Fredholm module (h, D) on a C* algebra A implies the existence of a trace state on A and that no such module exists on the C*…

Finitely summable Fredholm modules over higher rank groups and lattices

- Mathematics
- 2008

We give a complete classification (up to smooth homotopy) of finitely summable Fredholm representations (Fredholm modules) over higher rank groups and lattices. Our results are a direct consequence…

e Duality and Spectral Triples for Hyperbolic Dynamical Systems

- Mathematics
- 2010

We study aspects of noncommutative geometry on hyperbolic dynamical systems known as Smale spaces. In particular, there are two C∗-algebras, defined on the stable and unstable groupoids arising from…

C*-algebras and self-similar groups

- Mathematics
- 2009

Abstract We study Cuntz-Pimsner algebras naturally associated with self-similar groups (like iterated monodromy groups of expanding dynamical systems). In particular, we show how to reconstruct the…

K-homological finiteness and hyperbolic groups

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2018

Abstract Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C * \mathrm{C}^{*} -algebra has uniformly summable K-homology…

K-Theoretic Duality for Shifts of Finite Type

- Mathematics
- 1997

Abstract:We will study the stable and unstable Ruelle algebras associated to a hyperbolic homeomorphism of a compact space. To do this, we will describe a notion of K-theoretic duality for -algebras…

Spectral triples and finite summability on Cuntz-Krieger algebras

- Mathematics
- 2014

We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space"…

Nuclear dimension and classification of C*-algebras associated to Smale spaces

- Mathematics
- 2016

We show that the homoclinic C*-algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C*-algebras…

TOPOLOGICAL INVARIANTS OF ELLIPTIC OPERATORS. I: K-HOMOLOGY

- Mathematics
- 1975

In this paper the homological K-functor is defined on the category of involutory Banach algebras, and Bott periodicity is proved, along with a series of theorems corresponding to the…

The Structure ofC*-Algebras Associated with Hyperbolic Dynamical Systems☆☆☆

- Mathematics
- 1999

We consider the stable, unstableC*-algebras and the Ruelle algebras associated to a mixing Smale space. In the case of a shift of finite type, these are the AF-algebras studied by W. Krieger and the…