Conformal Markov Systems, Patterson-sullivan Measure on Limit Sets and Spectral Triples

One theme in noncommutative geometry is the recovery of classical geometric data from operator algebraic descriptions of various spaces, formalized through the concept of a spectral triple. This is defined below but, roughly speaking, consists of a C-algebra acting on a Hilbert space, giving co-ordinates for the space, and a Dirac operator, which encodes geometric information. As a prototypical example, Connes showed how to recover the volume measure and metric on a spin Riemannian manifold [11, 12]. As well as manifolds, many fractal sets may be described within this framework. Using ideas of Lapidus and Pomerance [20], Connes gave a construction of a spectral triple for a Cantor subset of the real line, from which the Minkowski content may be recovered, and used this to study the limit set of Fuchsian Schottky groups ([11], Chapter IV, §3.ǫ). Using a different approach, Connes and Sullivan constructed Patterson-Sullivan measue for a quasi-Fuchsian group as a noncommutative measure ([11], Chapter IV, §3.γ). Connes’s original approach was generalized and the subject considerably advanced in the last decade by Guido and Isola [15, 16, 17]. They defined spectral triples associated to fractal subsets of R obtained as a result of a limit construction. In particular, this includes the limit sets of finite state iterated function systems (IFSs). If the transformations are similarities (so the limit set is self-similar), the δ-dimensional Hausdorff measure, where δ is the Hausdorff dimension, can be recovered as a noncommutative measure. Similar results have been obtained for more general Gibbs measures by Kesseböhmer and Samuel [19, 28] and the author [31], where the underlying dynamics is an expanding map on a Cantor set or a (onesided) subshift of finite type. (These should be compared with the constructions in [10].) Falconer and Samuel [14] have begun to investigare multifractal phenomena for Cantor subsets of the real line. In the self-similar setting, the Hausdorff measure arises as a conformal measure and the recovery of conformal measures will be the subject of this paper. We will do this in the context of conformal graph directed Markov systems (CGDMSs) (defined in section 2), which provide a natural generalization of IFSs. In greater generality, Palmer [23], following work of Pearson and Bellissard [26], has shown how to construct spectral triples from which the Hausdorff measure (in the Hausdorff dimension) may be recovered. However, his construction is not explicit and, in the cases where the conformal measure is Hausdorff measure, our