We are concerned with the tiling flow T associated to a substitution φ over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue λ > 1. The motivation is provided by the (still open) conjecture asserting that T has pure discrete spectrum for any such φ. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual R d -1 -action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as sufficient conditions only-but they all emerge from a unified approach based on the new device: the strand space F φ of φ. The proof of the necessity hinges on determination of the discrete spectrum of T as that of the associated Kronecker toral flow.

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. The results of this paper relate the dynamics of a continuous map/of the interval and the topology of the inverse limit space with bonding map /. These inverse limit spaces have been studied by many authors, and are examples of what Bing has called "snakelike continua". Roughly speaking, we show that when the dynamics of / are complicated, the inverse limit space contains indecomposable subcontinua. We also establish a partial converse.

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.

We prove the Ingram conjecture, ie we show that the inverse limit spaces of tent maps with different slopes in the interval OE1; 2 are nonhomeomorphic. Based on the structure obtained from the proof, we also show that every self-homeomorphism of the inverse limit space of a tent map is pseudo-isotopic, on the core, to some power of the shift homeomorphism. 54H20; 37B45, 37E05

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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.

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