Asymptotic structure in substitution tiling spaces

作者：Barge Marcy^*; Olimb Carl

来源：Ergodic Theory and Dynamical Systems, 2014, 34: 55-94.

DOI：10.1017/etds.2012.118

Every sufficiently regular non-periodic space of tilings of R-d has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open (d - 1)-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity %26apos;starts%26apos;. This leads to the definition of the branch locus of the tiling space: this is a subspace of the tiling space, of dimension at most d - 1, that summarizes the %26apos;asymptotic in at least a half-space%26apos; behavior in the tiling space. We prove that if a d-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed (d - 1)-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a two-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a zero-or one-dimensional simplicial complex.

• 出版日期2014-2