Hyperbolic Dynamics and Noncommutative Geometry

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In the 1980s, Alain Connes, who had already won the Fields Medal for his work on C*-algebras, developed a new branch of mathematics called noncommutative geometry. Partly inspired by the description of subatomic phenomena given by quantum mechanics, the theory aimed to describe a wide variety of geometric objects in algebraic terms, where "points" are replaced by "operators". Connes was able to extend most of the tools of classical differential geometry to this setting but the theory is sufficiently flexible to allow a description of less regular objects: badly behaved quotient spaces, spaces of foliations and, particularly relevant to this application, fractal sets. Indeed, "fractal noncommutative geometry" has become a very active field in its own right.A very recent development has been the combination of fractal noncommutative geometry with the theory of hyperbolic dynamical systems. The latter are the prototypical examples of chaotic dynamical systems and are characterized by a local decomposition into exponentially expanding and contracting directions. They have a rich orbit structure and many important characteristics, for example invariant measures, can be recovered from averaging over families of orbits. Such families of orbits can also be used to construct to objects required for a noncommutative description of the geometry of the dynamical system and this is an aspect we intend to exploit.Our principle objective is to describe the invariant set of a hyperbolic dynamical system, together with an important class of invariant measures, called Gibbs measures, in terms of noncommutative geometry or, more technically, in terms of an object called a spectral triple. This includes an operator, called a Dirac operator, which provides the analogue of differentiation. We also aim to develop this theory for the limit sets of Kleinian groups, which can appear as intricate fractal patterns on the the two dimensional sphere.We further aim to develop a noncommutative, or spectral, theory of dynamics and Kleinian group actions. To this end, we will study spectral metric spaces associated to algebraic objects coming from the simplest Kleinian groups, namely Schottky groups. In these examples, the limit set is a Cantor set, one of the most familiar examples of fractal set. Spectral triples assiciated to Cantor sets have also been studied recently by Bellisard and Pearson and they were led to define a Laplace-Beltrami operator in this setting. We aim to extend this work to a wider setting.Finally, we aim to develop a mutifractal analysis -- the study of the fine fractal structure of dynamical systems -- in terms of noncommutative geometry.