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.