There are many ways of introducing quasiperiodicity into a model. 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). Trace map dynamics: general results with recent applications in the theory of orthogonal polynomials and classical Ising models. Read more about Trace map dynamics: general results with recent applications in the theory of orthogonal polynomials and classical Ising models (II).Time permitting, we'll also mention recent applications in the theory of orthogonal polynomials. 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. ![]() 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. Last time we saw how dynamical systems are associated to certain quasiperiodic models in physics. Trace map dynamics: general results with recent applications in the theory of orthogonal polynomials and classical Ising models (II) Read more about Trace map dynamics: general results with recent applications in the theory of orthogonal polynomials and classical Ising models (III).Should we have time, we'll also very briefly mention applications of the aforementioned dictionary to quasiperiodic Jacobi matrices/CMV matrices. The purpose of this work is to serve as rigorous justification to previously observed phenomena (mostly through numerical and some soft analysis). 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). 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 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. Advancement and Dissertation Guidelines.Course Registration and Placement Information.Calculus and Precalculus Student Enrollment Guide.Mathematics of Complex Social Phenomena.Analysis and Partial Differential Equations.Providence: American Mathematical Society. Ordinary Differential Equations and Dynamical Systems. An introduction to symbolic dynamics and coding. ISBN 0-19-853390-X (Provides a short expository introduction, with exercises, and extensive references.) Keane, Ergodic theory and subshifts of finite type, (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Natasha Jonoska, Subshifts of Finite Type, Sofic Systems and Graphs, (2000).Substitutions in dynamics, arithmetics and combinatorics. Berthé, Valérie Ferenczi, Sébastien Mauduit, Christian et al. ![]()
0 Comments
Leave a Reply. |