arXiv subjects
Publications and source records attributed to John Hunton.
We consider the ways minimal sets of flows in $S^3$ may be embedded. We prove that given any $C^2$ flow on $S^3$ with positive entropy, there is an uncountable collection $\mathcal{M}$ of topologically distinct minimal sets such that for each $M\in \mathcal{M}$ there are infinitely many embedded copies of $M$ in the flow, each copy with a distinct knot type, thus extending work of Franks and Williams for periodic orbits.
Minimal flow spaces of dimension 1 are among the most fundamental limit sets in dynamical systems. These invariant sets occur as the typical minimal sets in surface flows, the minimal sets of suspensions of subshifts (for example, in Lorenz template models of the Lorenz attractor) and the hulls of repetitive tilings of dimension one. Here we establish a complete invariant for the flow equivalence of such objects. The invariant takes values in a category of `positive trope' classes of inverse sequences of free groups and positive maps, or alternatively within a certain category of symbolic systems. Moreover, every such symbolic system is realised by a flow space, and we thus have a one to one correspondence between flow equivalence classes of minimal flow spaces and positive trope classes of such systems. At the same time, this provides a complete invariant both for flow equivalence of minimal Z-Cantor dynamical systems and for germinal equivalence of minimal Z-Cantor systems. Our work thus greatly extends that of Barge and Diamond on their complete invariant of primitive substitution tilings, and provides counterpoint to the work of Giordano, Putnam and Skau on orbit equivalence of Z-Cantor systems.
We present a definition of chaotic Delone set, and establish the genericity of chaos in the space of $(\epsilon,\delta)$-Delone sets for $\epsilon\geq \delta$. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.
We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structures, studying an associated space, the continuous hull, here denoted $\Omega_t$. In this article we consider two further spaces $\Omega_r$ and $\Omega_G$ (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. The rotational hull $\Omega_r$ is shown to be a matchbox manifold which contains $\Omega_t$ as a sub-matchbox manifold. We develop new S-MLD invariants derived from the homotopical and cohomological properties of these spaces demonstrating their computational as well as theoretical utility. We compute these invariants for a variety of examples, including a class of 3-dimensional aperiodic patterns, as well as for the space of periodic tessellations of $\mathbb{R}^3$ by unit cubes. We show that the classical space group of symmetries of a periodic pattern may be recovered as the fundamental group of our space $\Omega_G$. Similarly, for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.
Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological invariants related to their expansions as an inverse sequence of simplicial complexes. These invariants are related to corresponding inverse sequences of groups arising from applying the top--dimension homology to these sequences. In many cases this leads to a computable invariant based on an inverse sequence of matrices. Significantly, we show that when the space is obtained by suspending a topologically transitive action of the fundamental group $\G$ of a closed orientable on a zero--dimensional compact space this invariant at the same time corresponds to the space of Borel measures on the Cantor set which are invariant under the action of $\G$. This leads to connections between the rank of homology groups we consider and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.
The first author's recent unexpected discovery of torsion in the integral cohomology of the Tübingen Triangle Tiling has led to a re-evaluation of current descriptions of and calculational methods for the topological invariants associated with aperiodic tilings. The existence of torsion calls into question the previously assumed equivalence of cohomological and K-theoretic invariants as well as the supposed lack of torsion in the latter. In this paper we examine in detail the topological invariants of canonical projection tilings; we extend results of Forrest, Hunton and Kellendonk to give a full treatment of the torsion in the cohomology of such tilings in codimension at most 3, and present the additions and amendments needed to previous results and calculations in the literature. It is straightforward to give a complete treatment of the torsion components for tilings of codimension 1 and 2, but the case of codimension 3 is a good deal more complicated, and we illustrate our methods with the calculations of all four icosahedral tilings previously considered. Turning to the K-theoretic invariants, we show that cohomology and K-theory agree for all canonical projection tilings in (physical) dimension at most 3, thus proving the existence of torsion in, for example, the K-theory of the Tübingen Triangle Tiling. The question of the equivalence of cohomology and K-theory for tilings of higher dimensional euclidean space remains open.
We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomological invariants. Our framework unifies the several previous methods used to study the cohomology of these patterns. We discuss explicit calculations for the main examples of icosahedral patterns in $R^3$ -- the Danzer tiling, the Ammann-Kramer tiling and the Canonical and Dual Canonical $D_6$ tilings, including complete computations for the first of these, as well as results for many of the better known 2 dimensional examples.
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 space with a 2 to 1 covering by such a torus-less-points. Furthermore, we show that each orientable attractor is homeomorphic to a tiling space associated to an aperiodic tiling of Rd, but that the converse is generally not true. This work allows the definition of a new invariant for aperiodic tilings, in many cases finer than the cohomological or K-theoretic invariants studied to date.
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.
Aperiodic point sets (or tilings) which can be obtained by the method of cut and projection from higher dimensional periodic sets play an important role for the description of quasicrystals. Their topological invariants can be computed using the higher dimensional periodic structure. We report on the results obtained for the cohomology groups of projection point patterns supplemented by explicit calculations made by F. Gaehler for many well-known icosahedral tilings.
We analyze and compare different dynamical systems and groupoids which can be obtained from projection point patterns. We define the cohomology of a point pattern as the cocycle cohomology of the pattern groupoid. We describe this cohomology qualitatively and calculate it for canonical projection method tilings with codimension smaller or equal than 3.