2011/04/28 by Xifeng Su, Su, Xifeng, Rafael de la Llave +1
Materials Science · Physics and Astronomy · #37A60 #37J40 #52C23 #70K43 #82B20 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Quasicrystal Structures and Properties #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1104.5274
openalex publication_date 2011/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider Frenkel-Kontorova models corresponding to 1 dimensional quasicrystals. We present a KAM theory for quasi-periodic equilibria. The theorem presented has an a-posteriori format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then, there is a true solution nearby. This solution is locally unique. Such a-posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences a) Existence to all orders of perturbative expansion and their convergence b) Bootstrap for regularity c) An efficient method to compute the breakdown of analyticity. Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities. These identities also lead to very efficient algorithms.