The circle and the solenoid

In the paper, we discuss two questions about degree d smooth expanding circle maps, with d ≥ 2. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequences of asymptotic length ratios are precisely those given by a positive Hölder continuous function s (solenoid function) on the Cantor set C of d-adic integers satisfying a functional equation called the matching condition. In the case of the 2-adic integer Cantor set, the functional equation is s(2x +1) = s(x)/s(2x) 1 + 1/s(2x - 1) - 1. We also present a one-to-one correspondence between solenoid functions and affine classes of exponentially fast d-adic tilings of the real line that are fixed points of the d-amalgamation operator, (ii) We calculate the precise maximum possible level of smoothness for a representative of the system, up to diffeomorphic conjugacy, in terms of the functions s and cr(x) = (1 + s(x))/(1 + (s(x + 1))^-1). For example, in the Lipschitz structure on C determined by s, the maximum smoothness is C^1+α for 0 < α ≤ 1 if and only if s is α-Hölder continuous. The maximum smoothness is C^2+α for 0 < α ≤ 1 if and only if cr is (1 + α)-Hölder. A curious connection with Mostow type rigidity is provided by the fact that s must be constant if it is α-Hölder for a > 1.