The circle and the solenoid
This very interesting paper gives a comprehensive study of scaling and solenoid functions. These functions represent complete invariants of the smooth conjugacy classes of degree \(d\) smooth expanding circle maps (for \(d\geq 2\)). However, the two notions are not the same. Namely, a scaling function does not determine a moduli space for an expanding circle map, whereas a solenoid function does determine a moduli space. The main results of the paper are the following. (i) A characterization of the sequences of asymptotic length ratios which can occur for systems with Hölder continuous derivative. These ratios are precisely those given by a positive Hölder continuous function \(s\) (the solenoid function) on the Cantor set \(C\) of \(d\)-adic integers satisfying a certain matching condition. Moreover, it is shown that there is a one-to-one correspondence between solonoid functions and affine classes of exponentially fast \(d\)-tilings of the reals. (ii) The explicit calculation of the precise maximal possible level of smoothness for a representation of the expanding circle map, up to diffeomorphic conjugacy, in terms of the function \(s\) and a certain cross ratio function \(t\). For example, in the Lipschitz structure on \(C\) determined by \(s\), the maximal smoothness is \(C^{1+\alpha}\) for \(0<\alpha \leq 1\) if and only if \(s\) is \(\alpha\)-Hölder continuous. The maximal smoothness is \(C^{2+\alpha}\) for \(0<\alpha \leq 1\) if and only if \(t\) is \((1+\alpha)\)-Hölder continuous. An interesting connection with Mostow type rigidity is provided by the fact that \(s\) has to be constant if \(s\) is \(\alpha\)-Hölder continuous for \(\alpha >1\).
scientific article; zbMATH DE number 5121901