Circle maps driven by a class of uniformly distributed sequences on T$\mathbb {T}$

We use certain uniformly distributed sequences on T=R/Z$\mathbb {T}=\mathbb {R}/\mathbb {Z}$, including the sequence (n!xmod1)n > 0$(n!x \mod {1})_{n\geqslant 0}$, to drive families of circle maps. We show that: (1) under mild assumptions on the function v:T -> R$v:\mathbb {T}\rightarrow \mathbb {R}$, the discrete Schrodinger equation on the half line, with a potential of the form lambda v(n!x)$\lambda v(n!x)$ where lambda>1$\lambda >1$ is large, has for all energies E is an element of R$E\in \mathbb {R}$ exponentially growing solutions for almost every (a.e.) x is an element of T$x\in \mathbb {T}$; (2) the derivative of compositions fn circle fn-1 circle MIDLINE HORIZONTAL ELLIPSIS circle f1$f_n\circ f_{n-1}\circ \cdots \circ {f_1}$, where fj(t)=bt+(j!x)+b2 pi sin(2 pi t)mod1$f_j(t)=bt+(j!x)+\frac{b}{2\pi }\sin (2\pi t) \mod {1}$ (b is an element of Z,b > 3$b\in \mathbb {Z}, b\geqslant 3$) grow exponentially fast with n$n$ for a.e. x,t is an element of T$x,t\in \mathbb {T}$.