Dynamical Systems Seminar

Spectral triples were introduced by Connes as a generalization of a derivative to the setting of non-commutative geometry. In rough terms, a spectral triple consists of a Hilbert space \(H\), an unbounded, (essentially) selfadjoint operator \(D\) acting on \(H\) and a representation of a C^*-algebra \(A\) as bounded operators of \(H\). Having a spectral triple at hand, one may look at the Dixmier trace of the commutator of \(D\) and an element of \(A\) and refer to it as a non-commutative integral.

In this talk, a realization of a spectral triple in the context of a new class of weakly expanding maps is given. It then turns out that it is easy to construct a spectral triple but that it is less obvious if the topology generated by these triples coincides with the original one. Furthermore, it turns out that the non-commutative integral is a well-known object as it coincides in this setting essentially coincides with integration by the equilibrium state.

The proof of the second part is of independent interest as it relies on polynomial decay of correlations through a new Perron-Frobenius-Ruelle theorem for either expanding maps/irregular potentials or weakly expanding maps/Hölder potentials.

It is also worth noting that the approach does not make use of partitions. In particular, it is possible to handle weakly expanding maps on manifolds, pathwise connected fractals and shift spaces in a unified way.