Questions of continuity of the Lyapunov exponent play an important role in the spectral theory of quasi-periodic Jacobi matrices. Purpose of this talk is to present a survey of available positive and negative results for general, quasi-periodic M(2,C)-cocycles.
Motivated by the theory of fractal strings and complex dimensions of M. L. Lapidus and M. van Frankenhuijsen, we define a class of fractal strings for self-similar measures based on scaling regularity. In turn, these fractal strings allow for an analysis of the symbolic dynamics on such measures via the abscissae of convergence of scaling zeta functions. With this approach, we recover (among other things) Moran's theorem regarding the Hausdorff dimension of self-similar sets and the Hausdorff dimensions of Besicovitch subsets.
We shall review basic properties of cocycles over a minimal dynamical system, taking values in the special linear group of two by two matrices over the real numbers. It turns out that dynamical properties of such cocycles play a central role in the spectral theory of quasiperiodic one-dimensional Hamiltonians. We shall review those dynamical properties and connections with spectral theory. This talk will be of expository nature, and technical details will be kept to a minimum (respectively, we shall assume no prior background in the subject).
We will discuss several open problems on dynamical properties of interval translation mappings. In particular, we will observe and discuss an interesting change in the limiting behavior in the case when some randomness is added to the system.
Bruin-Troubetzkoy family is a two parameter family of interval translation mappings. It was studied using a renormalization procedure that provides a satisfactory description of its dynamical properties. In the talk we will describe this procedure, list the known properties of the family, and state some new results.
In the previous two talks we established a dictionary between some properties of quasiperiodic (particularly Fibonacci) models and some geometric constructions arising as dynamical invariants for the Fibonacci trace map. In this talk we shall apply our findings to a specific model: the classical 1D Ising model with quasiperiodic magnetic field and quasiperiodic nearest neighbor interaction. In particular, we'll prove absence of phase transitions of any order and we'll investigate the structure of Lee-Yang zeroes in the thermodynamic limit (these are zeroes of the partition function as a function of the complexified magnetic field---while in finite volume the partition function is a polynomial whose zeroes fall on the unit circle, a challenge is to determine whether in infinite volume (thermodynamic limit) these zeroes accumulate on any set on the unit circle, and if so, to determine the structure of this set). The purpose of this work is to serve as rigorous justification to previously observed phenomena (mostly through numerical and some soft analysis). Should we have time, we'll also very briefly mention applications of the aforementioned dictionary to quasiperiodic Jacobi matrices/CMV matrices.
Last time we saw how dynamical systems are associated to certain quasiperiodic models in physics. We also saw the need for a general investigation of dynamics of trace maps and the geometry of some dynamically invariant sets, motivating this week's discussion. We'll investigate in greater generality dynamics of the Fibonacci trace map, geometry of so-called stable manifolds, and we'll see how this information can be used to get detailed topological, measure-theoretic and fractal-dimensional description of spectra of quasiperiodic (Fibonacci) Schroedinger and Jacobi Hamiltonians, as well as the distribution of Lee-Yang zeros for the classical Ising model. Time permitting, we'll also mention recent applications in the theory of orthogonal polynomials.
Over the past almost three decades dynamical systems have played a central role in spectral analysis of quasiperiodic Hamiltonians as well as certain quasiperiodic models in statistical mechanics (most notably: the Ising model, both quantum and classical). There are many ways of introducing quasiperiodicity into a model. We shall concentrate on the widely studied Fibonacci case (which is a prototypical example of so-called substitution systems on two letters with certain desirable properties). In this case a particular geometric scheme, arising from a certain smooth three-dimensional dynamical system associated to the quasiperiodic model in question (the so-called Fibonacci trace map) has been established. Our aim is to present a general dynamical/geometric framework and to demonstrate how information about the model in question (spectral properties for Hamiltonians, and Lee-Yang zeros distribution for classical Ising models) can be obtained from the aforementioned dynamical system and the geometry of certain dynamically invariant sets. In this first in a series of two (or three) talks, we'll briefly recall how dynamical systems are associated to Schroedinger and Jacobi operators, as well as classical Ising models. We'll establish notation, ask main questions and in general prepare the ground for a somewhat more general (in terms of geometry and dynamical systems) discussion for next time.
We will discuss some properties of primitive, invertible substitutions and substitution sequences. Also, we will examine the relationship between such substitution sequences and Sturmian sequences.
Institute for Mathematics, Free University of Berlin
Time:
Friday, January 13, 2012 - 2:00pm
Location:
RH 440R
The shadowing problem is related to the following question: under which condition, for any pseudotrajectory approximate trajectory) of a vector field there exists a close trajectory? It is known that in a neighbourhood of a hyperbolic set diffeomorphisms and vector fields have shadowing property. In fact more general statement is correct: structurally stable dynamical systems satisfy shadowing property.
We are interested if converse implication is correct. We consider notion of Lipschitz shadowing property and proved that it is equivalent to structural stability for the cases of diffeomorphisms and vector fields.
Talk is based on joint works with S. Pilyugin and K. Palmer