Tiling groupoids and Bratteli diagrams

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths dB is homeomorphic to X. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an etale equivalence relation R_B on dB which is homeomorphic to R_X, and contains the AF-relation of "tail equivalence".Comment: 34 pages, 4 figure

\'Etale groupoids and Steinberg algebras, a concise introduction

We give a concise introduction to (discrete) algebras arising from \'etale groupoids, (aka Steinberg algebras) and describe their close relationship with groupoid C*-algebras. Their connection to partial group rings via inverse semigroups also explored

The 200 MeV Bremsstrahlung Tagged Photon Beam at LNS-Sendai(I. Nuclear Physics)

We describe the 200-MeV bremsstrahlung photon tagging system, which was installed in the experimental Hall-1 at the Laboratory of Nuclear Science, Tohoku University. This system produces tagged photons from high duty electron beams supplied by Stretcher Booster Ring. The tagged photon energies are over a range from 20% to 80% of the incident electron energy. We carried out commissioning to examine the performance of the tagged photon beams produced by the 198 MeV electron beam. We demonstrated that the tagged photons can be employed for photonuclear reaction experiments with a momentum resolution Δp/p～1% at the tagged photon intensity I≤5×10^6