Dynamical Systems Applied to Asymptotic Geometry

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 H\"older continuous derivative. The sequences of asymptotic length ratios are precisely those given by a positive H\"older 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 $s(2x+1)=\frac{s(x)}{s(2x)}(1+\frac{1}{s(2x-1)})-1.$ We also present a one-to-one correspondence between solenoid functions and affine classes of 2-adic quasiperiodic tilings of the real line that are fixed points of the 2-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 $C^{1+\alpha}$ for $0<\alpha\le 1$ if, and only if, $s$ is $\alpha$-H\"older continuous. The maximum smoothness is $C^{2+\alpha}$ for $0<\alpha\le 1$ if, and only if, $cr$ is $(1+\alpha)$-H\"older. A curious connection with Mostow type rigidity is provided by the fact that $s$ must be constant if it is $\alpha$-H\"older for $\alpha > 1$.