In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{cc}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)\hfill & 0\<x\<\pi ,t\in ℝ,i\in \mathbb{Z}\left(TC\right)\hfill \\ {\varphi }^i(0,t)={\varphi }^i(\pi ,t)=0\hfill & \forall t,i.\hfill \end{array}\right.$$We consider the case of “closed chains” i.e. ${\varphi }^{i+N}={\varphi }^i\forall i\in \mathbb{Z}$ and some $N\in \mathbb{N}$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{c}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)0<x<\pi ,t\in \mathrm{I}\mathrm{R},\mathrm{i}\in \mathrm{Z}\mathrm{Z}\left(\mathrm{TC}\right){\varphi }^{\mathrm{i}}(0,\mathrm{t})={\varphi }^{\mathrm{i}}(\pi ,\mathrm{t})=0\forall \mathrm{t},\mathrm{i}.\hfill \end{array}\right.$$ We consider the case of “closed chains" i.e.${\varphi }^{i+N}={\varphi }^i\forall i\in ZZ$ and some $N\in IN$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap,...