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.

This is the first in a series of two (or three) talks on partial (and normal) hyperbolicity. Partial hyperbolicity is in a sence a generalization of the notion of uniform hyperbolicity -- a well developed branch of smooth dynamical systems. In this talk we will begin with a motivation, definitions and some basic examples, laying the ground for the subsequent discussion of more advanced topics (mainly questions concerning generalization of resulrts of hyperbolic dynamics to partially hyperbolic systems).