Shift radix system (SRS) is a simple contractive algorithm acting on a lattice. This has interesting connection to several well-known numeration systems. We shall try to describe in a primitive manner, the idea used to show finiteness and periodicity of this system. Hopefully we can also discuss several open questions on it. Especially we present a proof of `non-finiteness' result on the boundary of 3-dim SRS.
In the talk we discuss Barry Simon's subshift conjecture, which states that the (OPUC or Schrödinger) spectrum associated with a minimal aperiodic subshift has zero Lebesgue measure. We use approximation with subshifts of finite type along with an analysis of the spectra associated with periodic orbits in them to produce a counterexample. This is joint work with Artur Avila and Zhenghe Zhang.
In the talk I will describe recent and ongoing work with Margulis and Venkatesh which concerns the error rate of the equidistribution of finite volume orbits xH in homogeneous spaces X=Gamma \ G. Mozes and Shah have shown (building on the measure classification theorem of Ratner) that a weak* limit of probability measures that are invariant and ergodic under a unipotent flow (Haar measures by Ratner's theorem) is either 0 (complete escape of mass) or again the Haar probability measures on a closed orbit (equidistribution). However, in general their proof does not provide an error rate. Much earlier, Sarnak was able to give an error rate for the equidistribution of closed horocycle orbits on non-compact quotients of SL_2(R). I will explain the theorem with Margulis and Venkatesh which gives an error rate for the equidistribution of Haar measures of closed orbits of certain semi-simple subgroups H < G. The proof uses ideas from the measure classification theorem, effective decay of matrix coefficients, and property (tau). Time permitting I will also outline how the method can also be used to prove a weak form of property (tau).
The study of dynamical zeta functions is a part of the theory of dynamical systems, but it also intimately related to algebraic geometry, number theory, topology and statistical mechanics. In the talk I will discuss zeta functions which count periodic points of dynamical system in the presence of fundamental group. Arithmetical congruences for Reidemeister numbers will be described. I will explain how dynamical zeta functions give rise to the Reidemeister torsion, a important topological invariant. The connection between symplectic Floer homology for surfaces, topological entropy and Nielsen fixed point theory will be described. In the talk a categorification of Weil type dynamical zeta functions is proposed.
There exists a close relation between the dynamical zeta function of the substitution dynamical system of a substitution tiling, and the substitution action on the Cech cohomology groups of its tiling space. A link ist thereby obtained between the periodic orbits under the substitution, which determine the zeta function, with the topology of the tiling space. After a brief review of the theory, we illustrate this relation with a number of examples, notably some lattice substitution tilings, such as the chair tiling, the squiral tiling, and some of their factors and relatives. It turns out that the zeta function is a very helpful tool to deduce the structure of a tiling space.
The Fibonacci Trace Map $T:(x, y, z)\mapsto (2xy-z, x, y)$ appears in a natural way in connection with many questions in mathematical physics, analysis, and number theory. In particular, it appears in several problems related to spectral properties of Fibinacci quasicrystals. It has a first integral (the Fricke-Vogt invariant), and restriction of $T$ to the level surfaces that correspond to positive values of the invariant is hyperbolic. The set of periodic orbits in this case (all of them are hyperbolic saddles) can be explicitly described. On the other hand, on the level surfaces that correspond to negative values of the Fricke-Vogt invariant the map $T$ has both hyperbolic and elliptic periodic orbits, and we show that for many values of the first integral the number of elliptic periodic orbits is also infinite (conservative Newhouse phenomenon), the dynamics is not structurally stable, and in many ways is similar to the dynamics of the famous standard family. We will also formulate some open questions on dynamical properties of the Trace Map.
The squiral tiling is a planar inflation tiling. Its inflation rule is equivalent to a simple bijective block substitution rule. This leads to a lattice dynamical system with interesting properties. In particular, it can be viewed as a two-dimensional generalisation of the Thue-Morse system, and for balanced weights shows purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive approach that admits a generalisation to a large class of bijective block substitutions in any dimension.
Suppose we have a compact differentiable manifold M with a diffeomorphism d:M→M, defining a hyperbolic dynamical system. Suppose A is an attractor for this system: what can the space A look like, even through the eyes of basic topological invariants, cohomology, K-theory etc? Of course A can be very complicated indeed, but I will look at the case of those attractors of dimension one less than that of M, and show that there is a nice connection between each such attractor and certain moduli spaces of aperiodic tilings, objects for which there is already powerful machinery available for their analysis. Conversely, we derive constraints and obstructions for when a tiling space has a codimension-one embedding in a manifold. This is joint work with Alex Clark.
In joint work with Klaus Schmidt and Evgeny Verbitskiy, we investigate the problem of whether the growth rate of the number of periodic points for an algebraic action of Z^d exists and equals the entropy. Our (partial) solution involves the construction of well-behaved homoclinic points for the action, with excursions into number theory, real algebraic geometry, several complex variables, and even a bit of logic. However, some simple actions still remain beyond reach. Surprisingly, our approach also yields new Diophantine information. This study is part of a larger set of questions involving algebraic actions of discrete, residually finite groups.
Model sets have proven to be a significant source of aperiodic structures with long-range order -- in fact pure point diffractive order -- and this includes many of the famous aperiodic tilings. Our present understanding of model sets owes much to being able to put them into the context of dynamical systems. In this talk we review the role of dynamics in the theory of model sets and some of the achievements in the subject over the past 15 years. As a running example, we shall use the recently discovered Taylor aperiodic hexagonal tilings, in which the dynamics of the tiling hull can be made rather explicit.
Endomorphisms of the d-dimensional torus are induced by integer matrices M, which leave each rational lattice L_n (the set of n-division points) invariant. If the determinant of M shares a divisor with n, the action of M on L_n is not invertible and decomposes the lattice into orbits with `pretails' attached to each periodic point. In this talk, I will discuss the pretail-orbit structure of toral endomorphisms by characterising their induced graphs on the rational lattices L_n. This is joint work with Michael Baake and John Roberts.
Let $K$ be a solenoid and $\Ga$ be a group of automorphisms of $K$. We present conditions on $\Ga$ under which the semidirect product of $\Ga$ and $K$ has strong realtive property and the action of $\Ga$ on $K$ has spectral gap. This provides some interesting alternatives for $\Ga$.
The gamma distribution R(x)=1-exp(-x)(1+x) was previously found to describe the distribution of cycle lengths in permutations arising from the reduction to finite fields of rational automorphisms with a single time-reversal symmetry. In an attempt to investigate the minimal randomness needed to obtain R(x) in the reduction of a time-reversible map, we have studied numerically the Casati-Prosen map. This is a two-parameter reversible map of the torus with zero entropy which preserves rational lattices for rational parameter values. We consider the distribution of the periods over prime lattices and its dependence on the parameters of the map. We conjecture that, for a set of rational parameters having full density, the distribution converges asymptotically to the gamma distribution R(x). We show the non-uniformity of this convergence, a key feature being that R(x) emerges as the limit of a sequence of singular distributions on certain lines in the parameter space. This is joint work with Natascha Neumärker and Franco Vivaldi.
The Kari-Culik tilings provide a class of examples of Z^2 subshifts of finite type (or equivalently, aperiodic tilings) with no periodic points. These examples seem very different from most other examples, essentially all of which are produced by substitution. An important part of the argument -- that a Kari-Culik shift is nonempty -- displays an example of a single point in the shift. In this lecture we discuss a technique that displays a large set of these points, which we call the core. We study the properties of this core.
Naturally associated to a (compact) negatively curved Riemannian manifold is its length spectrum, i.e. the set of lengths of closed geodesics. This has been the subject considerable study using techniques from both spectral theory (in the case of constant curvature) and ergodic theory. We will discuss "pair correlations" within this spectrum: asymptotics for pairs of closed geodesics, the difference of whose length lies in some (possibly shrinking) interval. It is hoped that these type of results may shed some light on pair correlation problems in quantum chaos.
Quasicrystals are known for their peculiar properties and it is generally accepted that certain strange phenomena should occur in the spectral decomposition of the respective Laplacians. These are modeled with the help of certain dynamical systems. The talk is based on joint work with S. Klassert, D. Lenz and C. Seifert and gives an introductory account of the rigorous results and the open questions in this area of Mathematical Physics.
It is well-known that one can associate a Rauzy fractal with a given unit Pisot substitution. Recently, there were many attempts to generalize this concept in various ways. Together with V. Berthe and W. Steiner we currently study "S-adic" Rauzy fractals. These fractals are defined in terms of an infinite word that is generated by repeatedly applying substitutions (over the alphabet {1,..,n}) taken from a finite set S={sigma_1,..,sigma_k} to some starting letter. If the substitutions (and the way they are applied) satisfy certain properties we are able to show that the resulting Rauzy fractal is a non-empty compact set that induces a tiling of a certain hyperplane of the n-dimensional real vector space. In this talk we survey the results on this new kind of Rauzy fractals we got so far.
We consider a model of planar rotations subject to round-off, which leads to dynamics on a lattice. We let the angle of rotation approach a low-order rational. There is a non-smooth integrable Hamiltonian system, featuring a foliation by polygonal invariant curves, which represents the limit of vanishing discretisation of the space. We prove that, for sufficiently small discretization, a positive fraction of those invariant curves survives, leading to a discrete space version of the KAM scenario. The surviving curves are characterised in terms of congruences and properties of Gaussian integers.