Dynamical Methods in One-Dimensional Quasi-Periodic Lattice Models

Since the discovery by D. Shechtman in the early 1980's of the material that has become known as quasicrystal, mathematical models thereof have formed an active area of research. Quasicrystals are somewhat intermediate between crystals and amorphous matter (the former exhibiting periodic microscopic organization, while the latter does not exhibit any order at all). Quasicrystals possess a number of interesting properties, including quantum mechanical phenomena that had previously been regarded pathological. For this reason, mathematical models of quasicrystals and investigation thereof have occupied a center stage in mathematical physics (in particular in spectral theory) for the past thirty years. Some of the most widely research models of quasicrystals are the one-dimensional Schrödinger operator and the one-dimensional classical and quantum Ising models where, respectively, the potential and the neighbor interaction couplings are modulated by a sequence that is chosen to reflect the long-range (while still not periodic) order of the microscopic structure of the quasicrystals. In this dissertation we apply tools from ergodic theory and smooth and holomorphic dynamical systems to the spectral theory of one-dimensional classical and quantum Ising models and Jacobi operators (a generalization of the Schrödinger operator) on a one-dimensional quasi-periodic lattice, and Cantero-Moral-Velásquez matrices formed with a choice of quasi-periodic Verblunsky coefficients. We also present a canonical transformation from the quantum Ising model to the Jacobi operator (which has been known for a few decades) and from the classical Ising model to the Cantero-Moral-Velásquez matrix (which, to the best of our knowledge, is new). We model quasi-periodicity by the so-called Fibonacci substitution, though our methods can be generalized to a wider family of quasi-periodic sequences (this is discussed in some detail in the text). The highlight of this dissertation is a proof of multifractal structure of the spectrum of the aforementioned operators and investigation of its qualitative properties. By employing techniques from holomorphic dynamical systems, we also demonstrate rigorously absence of phase transitions of any order in the classical one-dimensional quasi-periodic nearest-neighbor Ising model with quasi-periodic magnetic field.