"Transverse measures in the theory of tiling spaces"
Hartnick, TobiasWe explain how Connes' theory of transverse measures can be used in the study of tiling spaces, both in Euclidean and non-Euclidean geometries. We will explain the general formalism and relate it to a number of classical subjects such as pattern equivariant functions, patch frequencies, densities of sphere packings and harmonic functions on pattern trees. In the cut-and-project case we explain how patch frequencies can be related to Haar measures of acceptance domains in the sense of Koivusalo-Walton through the theory of transverse measures. In the case of self-similar tilings we relate transverse measures to substitution matrices. In both cases, we provide new explicit computations of patch frequencies. Based on joint work with Michael Björklund, Yakov Karasik, Daniel Roca Gonzalez and Maximilian Wackenhuth.