Related papers: The Branch Locus for One-Dimensional Pisot Tiling …
Every sufficiently regular 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…
A substitution $\vp$ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot $\vp$ that satisfies a no cycle condition and for…
We show that any codimension one hyperbolic attractor of a diffeomorphism of a (d+1)-dimensional closed manifold is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the non-orientable case, to a…
Given an n-dimensional substitution whose associated linear expansion is unimodular and hyperbolic, we use elements of the one-dimensional integer \v{C}ech cohomology of the associated tiling space to construct a finite-to-one…
Using the notion of short directed path, we introduce the branching space of a multipointed $d$-space. We prove that for any q-cofibrant multipointed $d$-space, it is homeomorphic to the branching space of the q-cofibrant flow obtained by…
We consider the structure of Pisot substitution tiling spaces, in particular, the structure of those spaces for which the translation action does not have pure discrete spectrum. Such a space is always a measurable m-to-one cover of an…
We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space…
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…
Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f…
We consider 1-dimensional, unimodular Pisot substitution tilings with three intervals, and discuss conditions under which pairs of such tilings are locally isomorhphic (LI), or mutually locally derivable (MDL). For this purpose, we regard…
We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the…
Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained…
We give an application of a topological dynamics version of multidimensional Brown's lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$, one can find a patch such that, for each scale…
For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $\beta$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a…
In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2-dimensional SFTs (i.e. tilings). To be more precise, we prove that given any \pizu subset $P$ of $\{0,1\}^\NN$ there is a SFT $X$…
We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot"…
The main source of inspiration for the present paper is the work of R. Rosebrugh and R.J. Wood on constructive complete distributive lattices where the authors employ elegantly the concepts of adjunction and module in their study of ordered…
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a…
The quotient cohomology of tiling spaces is a topological invariant that relates a tiling space to one of its factors, viewed as topological dynamical systems. In particular, it is a relative version of the tiling cohomology that…
Let $\mathcal{M}_{g}$ be the moduli space of compact connected hyperbolic surfaces of genus $g\geq2$, and ${\mathcal B}_g \subset {\mathcal M}_{g} $ its branch locus. Let $\widehat{{\mathcal{M}}_{g}}$ be the Deligne-Mumford compactification…