Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.
Abstract. -Let ϕ be a substitution of Pisot type on the alphabet A = {1, 2, . . . , d}; ϕ satisfies the strong coincidence condition if for every i, j ∈ A, there are integers k, n such that ϕ n (i) and ϕ n (j) have the same k-th letter, and the prefixes of length k − 1 of ϕ n (i) and ϕ n (j) have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if d = 2 and provide a partial result for d ≥ 2.Résumé (Coïncidence pour les substitutions de type Pisot). -Soit ϕ une substitution de type Pisot sur un alphabet A = {1, 2, . . . , d} ; on dit que ϕ satisfait la condition de coïncidence forte si pour tout i, j ∈ A, il existe des entiers k, n tels que ϕ n (i) et ϕ n (j) aient la même k-ième lettre et les préfixes de longueur k − 1 de ϕ n (i) et ϕ n (j) aient la même image par l'application d'abélianisation. Nous montrons que la condition de coïncidence forte est satisfaite pour d = 2 et nous donnons un résultat partiel pour d ≥ 2.
Let \varphi be a primitive, non-periodic substitution. The tiling space \mathcal{T}_\varphi has a finite (non-zero) number of asymptotic composants. We describe the form and make use of these asymptotic composants to define a closely related substitution \varphi^* and prove that for primitive, non-periodic substitutions \varphi and \chi, \mathcal{T}_\varphi and \mathcal{T}_\chi are homeomorphic if and only if {\varphi^*} (or its reverse) and \chi^* are weakly equivalent. We also provide examples indicating that for substitution minimal systems, flow equivalence and orbit equivalence are independent.
Abstract. Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which "forces its border." One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. For one-dimensional substitution tiling spaces, we describe a modification of the Anderson-Putnam complex on collared tiles that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology.
Abstract. Taking inverse limits of the one-parameter family of tent maps of the interval generates a one-parameter family of inverse limit spaces. We prove that, for a dense set of parameters, these spaces are locally, at most points, the product of a Cantor set and an arc. On the other hand, we show that there is a dense G δ set of parameters for which the corresponding space has the property that each neighborhood in the space contains homeomorphic copies of every inverse limit of a tent map.In 1967, R. F. Williams ([10]) proved that hyperbolic one-dimensional attractors are inverse limits of maps on branched one-manifolds. These attractors have the solenoid-like property of being everywhere locally homeomorphic with the product of a Cantor set and an arc. Also, for dissipation parameter near zero, most of the full attracting sets for maps in the Hénon family are homeomorphic with inverse limits of unimodal maps of the interval ([1]). Except at finitely many points (the points of a stable periodic orbit), these sets are locally homeomorphic with the product of a Cantor set and an arc (see the comment following Theorem 1).Computer-generated pictures, at first glance, suggest that other one-dimensional (but non-hyperbolic) attractors might have a similar local structure. In particular, the transitive Hénon attractors appear to be, at most points, locally the product of a Cantor set and an arc. However, 'blowing up' computer pictures of these attractors usually indicates the presence of 'hooks' in the midst of regions that, under less scrutiny, look like a Cantor set of nearly parallel arcs.In this paper we consider the local topological properties of a one-parameter family of conceptual models for the Hénon attractors, inverse limits of tent maps. We find the following: for a dense set of parameters, the inverse limit space is, except at finitely many points, the product of a Cantor set and an arc (Theorem 1). However, for a dense G δ set of parameters, the inverse limit space is nowhere locally homeomorphic with the product of a Cantor set and an arc. In this second case, the inverse limit spaces display a remarkable form of self-similarity and local recapitulation of the entire family: not only does every open set in each space contain a homeomorphic copy of the entire space, each open set also contains a homeomorphic copy of every other inverse limit space appearing in the tent family (Corollary 6). In a forthcoming paper, we prove that the set of parameters for which this holds has full measure.
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