Niveau: Supérieur, Doctorat, Bac+8
PATTERN EQUIVARIANT FUNCTIONS, DEFORMATIONS AND EQUIVALENCE OF TILING SPACES JOHANNES KELLENDONK Abstract. We reinvestigate the theory of deformations of tilings using P -equivariant cohomology. In particular we relate the notion of asymptotically negligible shape functions introduced by Clark and Sadun to weakly P -equivariant forms. We then investigate more closely the relation between deformations of patterns and homeomorphism or topological conjugacy of pattern spaces. 1. Introduction The study of aperiodic systems in physics or geometry has led to the definition of cohomology groups associated with aperiodic tilings or point sets of Rn (we use here the word pattern to mean either of them) In physics some elements of these groups are related (via K-theory and cyclic cohomology) to topologically quantized transport properties, see [KR06] for a recent overview. In geometry Sadun, Williams and Clark have given an interpretation of the cohomology group (with values in Rn) in terms of deformation theory of tilings [SW03, CS06]. In short, an (admissible) 1-cocycle defines a deformation of a tiling by redefining the shape of its tiles. If the cocycle is a coboundary then the deformed tiling is locally derivable from the original one. A deformation alters the properties of the dynamical system associated with the tiling, except if the new tiling is topologically conjugate to the old one, a notion which is, however, strictly weaker than being mutually locally derivable.
- open ?- ball around
- rham cohomology
- implies relation
- such cocycles
- pattern equivariant
- converse can
- finite local