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.

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.

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

We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition. Thermostat is our prototype model.

By reducing the problem to a discontinuous infinite dynamical system, we construct all periodic solutions with exactly two switchings on the period and study their stability. In the problem under consideration, the hysteresis gap (the difference between the switching temperatures) is of especial importance.

If the hysteresis gap is large enough, then the constructed periodic solution is in fact unique and globally stable. For small values of hysteresis gap coexistence of several periodic solutions with different stability properties is proved to be possible.

This is a joint work with Pavel Gurevich.

In this talk we present a survey of our results on quasi-periodic Jacobi operators whose diagonal and off-diagonal elements are generated from two analytic functions on the circle. Such operators arise as effective Hamiltonians describing the effects of external magnetic fields on a tight binding, infinite crystal layer. The main motivation of our investigations was extended Harper's model (EHM), whose description on both the level of spectral analysis, as well the Lyapunov exponent (LE) had posed an open problem even from the point of view of physics literature. Among the topics that will be addressed are: Singular components of spectral measures for ergodic Jacobi operators, Singular analytic cocycles and joint continuity of the Lyapunov exponent, Recovery of spectral data from rational frequency approximants, Almost constant cocycles and the complexified LE of EHM, Spectral theory of EHM.