Anti-integrable limits and phonon gaps for generalised Frenkel-Kontorova models on almost-periodic media

I will discuss equilibrium configurations of generalised Frenkel-Kontorova models on almost-periodic media. In joint work with Xifeng Su, we establish locally unique equilibria with prescribed rotation data via anti-integrable limits for sufficiently strong substrate potentials, and study their hyperbolicity. I will then present results of Yujia An and Xifeng Su showing that, under the respective hypotheses, KAM equilibria have no phonon gap, whereas equilibria constructed via anti-integrable limits do.