We use certain uniformly distributed sequences on (Formula presented.), including the sequence (Formula presented.), to drive families of circle maps. We show that: (1) under mild assumptions on the function (Formula presented.), the discrete Schrödinger equation on the half line, with a potential of the form (Formula presented.) where (Formula presented.) is large, has for all energies (Formula presented.) exponentially growing solutions for almost every (a.e.) (Formula presented.); (2) the derivative of compositions (Formula presented.), where (Formula presented.) ((Formula presented.)) grow exponentially fast with (Formula presented.) for a.e. (Formula presented.).