In the paper we discuss two questions about smooth expanding dynamical systems on the circle. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function s (solenoid function) on the Cantor set C of 2-adic integers satisfying a functional equation called the matching condition. The functional equation for the 2-adic integer Cantor set is s(2x + 1) = s(x) s(2x)