Michael F. Whittaker

EQUILIBRIUM STATES ON THE CUNTZ-PIMSNER ALGEBRAS OF SELF-SIMILAR ACTIONS

We consider a family of Cuntz–Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz–Pimsner algebra; if the self-similar group is contracting, then the Cuntz–Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumotos notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their

Spectral triples for hyperbolic dynamical systems

C^*

Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs

-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs

Fractal dual substitution tilings

E

Functorial properties of Putnam's homology theory for Smale spaces

we construct a groupoid

Cohomology of the pinwheel tiling

\mathcal{G}_{(C^*(E),\mathcal{D}(E))}

Wieler solenoids, Cuntz–Pimsner algebras and K-theory

from the graph algebra

Zappa-Szép product groupoids and C∗-blends

C^*(E)

Topological spaces associated to higher-rank graphs

and its diagonal subalgebra