# Trace map dynamics: general results with recent applications in the theory of orthogonal polynomials and classical Ising models (III)

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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.