The wavevector-dependent susceptibility is the Fourier transform of the pair-correlation function revealing information on the structure of the model. The pair correlations in a planar Ising model can be calculated using quadratic recursion formulae. Aperiodicity can be introduced using Baxter's "Z-Invariance." We shall also introduce Ising models with aperiodic couplings based on de Bruijn's extension of the Fibonacci sequence. We shall prove an extension of a theorem of Tracy on the probability distribution implied and show some of our results.
An Ising model on a Penrose tiling based on a pentagrid is introduced. The Ising spins are on every other vertex of the Penrose tiling and the grid lines of the pentagrid are where the Yang-Baxter spectral parameters (rapidities) live. The pair correlation can be calculated by the techniques discussed in the previous talk. The joint probability for averaging over the two local environments of spins is reduced to linear programming. Finally results are shown for the wavevector-dependent susceptibility.
We present a new approach to quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators which is based on power-law bounds on transfer matrices. This approach is particularly useful for models which exhibit a small number of critical energies, such as polymer models, the period doubling Hamiltonian, or the Thue-Morse Hamiltonian. This is joint work with Serguei Tcheremchantsev and, in part, with András Sütõ.
Ammann tiles, Ammann bars -- Robert Ammann's many contributions to the theory of aperiodic tilings were ingenious, unexpected, and profound. Yet for the last 14 or so years of his life (1946 - 1994) he earned his living sorting mail in a Boston post office. In this talk I attempt to fit together the puzzling pieces of his unusual life and work.
We consider the R-action corresponding to a substitution tiling (or Delone set) on the line. In the measure-theoretic category, it is never strongly mixing. But what about the topological category? In the 2-symbol case, if both of the eigenvalues of the substitution matrix are greater than 1 in modulus and a certain combinatorial condition (e.g. the strong coincidence condition) holds, then the system is topologically mixing. One can then make higher-dimensional examples of topologically mixing systems by taking direct products. This is a preliminary report on a joint work with Rick Kenyon.
We consider tilings in d dimensions and the action of the translation group on these tilings. Deforming the shapes of the tiles, while leaving the combinatorics fixed, creates a new tiling space that is homeomorphic to the original one, but on which the translation group acts differently. The basic question is how such deformations affect the dynamics. We construct a map from the space of shape parameters to H1 of the tiling space with values in Rd and show that the dynamical properties of the tiling space depend only on the image of this map.
Symmetric polyhedra have been investigated since antiquity. With the passage of time, the concept of a polyhedron has undergone a number of changes which have brought to light new classes of regular polyhedra. Coxeter's famous "Regular Polytopes" and his various other writings treat the Platonic solids, the Kepler-Poinsot polyhedra and the Petrie-Coxeter in great detail, and cover what might be called the classical theory. A lot has happened in this area since then. In particular, in ordinary Euclidean 3-space, the class of regular polyhedra has been considerably extended to what are called the Grunbaum-Dress polyhedra. These are discrete polyhedral structures with finite or infinite, planar or skew, polygonal faces or vertex-figures. The talk presents a new approach to these polyhedra, which leads to a quicker proof of the completeness of the enumeration than that found by Dress, as well as to nice presentations of their symmetry groups. This is based on joint work with Peter McMullen.
The talk describes the complete enumeration of chiral or regular polyhedra in ordinary Euclidean 3-space. Chiral, or irreflexibly regular, polyhedra are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags (regular polyhedra have just one), such that adjacent flags are in distinct orbits. The classification of the regular polyhedra in 3-space was obtained by Grunbaum and Dress around 1980. However, the corresponding classification of the chiral polyhedra is new. There are several infinite families of chiral polyhedra, each with finite skew, or infinite helical, faces, and with finite skew vertex-figures. Their geometry and combinatorics are rather complicated.
A set T is called a multiplicative tile of R if there exists a subset A of R such that {aT: a in A} is a partition of R. In this talk I'll discuss the structure of multiplicative tiles and their tilings.
Let β>1 be noninteger. A β-expansion of x>=0 is any series in powers of 1/β with coefficients 0, 1, ..., [β] whose sum is equal to x. We will discuss the following problems:
1) "How many" β-expansions does x typically have?
2) Are there any x's that have a unique β-expansion; if so, how many?
3) What is the dynamical interpretation of β-expansions?
4) What can be said about the complexity of β-expansions?
Besides, we will present a new approach to constructing Sierpinski type fractals via two-dimensional β-expansions with "holes" and "overlaps".
We consider certain topological dynamical systems arising from translation bounded measures. To such a system we associate an autocorrelation measure. Given ergodicity, this autocorrelation measure agrees with "the usual one" obtained by a limiting procedure. In any case, its Fourier transform (diffraction measure) is shown to be a pure point measure if and only if the original dynamical system has pure points spectrum. (Joint work with Michael Baake.)
We study basic spectral features of random operators associated to Delone dynamical systems. For stricly ergodic systems we show existence of the integrated density of states (ids) in a very uniform sense. This is then used to characterize the points of discontinuity in the ids by means of locally supported eigenfunctions. For a class of continuum models based on the set of all Delone sets to certain parameters, we show Baire generic occurrence of purely singularly continous spectrum in a certain interval. (Joint work with Steffen Klassert and Peter Stollmann.)
We discuss the existence and origin of substitution rules for canonical projection tilings, including the 1D Fibonacci, and the 2D Ammann-Beenker and Penrose tiling. This is joint work with Edmund Harriss.
The presentation starts with an introduction in the theory of almost periodic measures. It continues with some applications to the diffraction, particularly to the diffraction of Meyer sets. The main result is that any Meyer set with a well defined autocorrelation has a diffraction pattern with a relatively dense set of Bragg peaks.
The shelling structure of a point set in Euclidean space consists of the number of points that lie on shells around an arbitrary, but fixed centre. For non-periodic point sets, where the shelling structure depends on the chosen centre, a more adequate quantity is the so-called averaged shelling, which is obtained by taking the average with all points in the set as centres. This radial distribution function is a characteristic geometric quantity that reflects itself in the corresponding diffraction spectrum and related objects of physical interest. The underlying combinatorial and algebraic structure is well understood for periodic crystals, but less so for non-periodic arrangements such as mathematical quasicrystals or model sets. In this talk, we summarise recent results on the averaged shelling, with particular emphasis on the example of planar model sets. In this case, the answer consists of a universal part that encodes properties of the underlying cyclotomic number field, and a non-universal part that depends on the details of the model set construction.
The hull of a repetitive tiling can be constructed as the inverse limit of a sequence of simpler spaces and continuous maps between them. These spaces, which approximate the hull, are determined by local patches of increasing size in the tiling. The inverse limit structure allows to compute the cohomology of the hull as the direct limit of the cohomologies of the approximating spaces. In favourite cases, the limit of the cohomologies is reached already after a finite number of steps, which not only makes the limit computable, but also expresses the cohomology in terms of the local structue of the tiling. The criteria for this to happen are discussed and elucidated with the help of several examples. The mechanism at work is similar to the one which makes matching rules be of finite range.
There is a criterion which gives a condition under which substitution point sets on a lattice become regular model sets. As an extension of this to general substitution getting away from the lattice situation, we will talk about a sufficient condition for general substitution point sets to be model sets.
The closely related topological invariants of K-theory and cohomology that can be associated to a tiling or pattern provide tools for discussing various phenomena such as aspects of the quantum mechanics of a corresponding crystal or the existence of substitution rules for a canonical projection tiling. This talk discusses these invariants and how they may in certain cases be computed in terms of invariants of Cantor dynamical systems. The talk will concentrate specifically on certain low dimensional canonical projection patterns.
The question, whether they admit quasi-periodic tilings, is still open. A tile-inflation cannot exist; at least not as long as we have no essentially generalised definition.
We begin with an introduction into the realm of automatic subsets of Zn. Furthermore, we present a necessary and sufficient condition for an automatic subset to be a Delone set.
In the past decade, there has been intense interest in integrable systems where time is discrete, i.e. integrable difference equations and integrable maps. We present a new method for testing integrability in rational maps of the real plane, by representing these maps over finite fields and examining their orbit structure. We find markedly different orbit statistics depending upon whether the map is integrable or not.