Robust rigidity for circle diffeomorphisms with singularities (Q948623)

The main result of this paper is the following: If \(T_1\) and \(T_2\) are two analytic critical maps of the circle to itself with critical points of the same order and the same irrational rotation number, then \(T_1\) and \(T_2\) are \(C^1\)-smoothly conjugate to each other. (A critical map is a homeomorphism of the circle which is smooth everywhere, except at one point where the first-derivative vanishes.) Yoccoz had originally proved topological equivalence under similar assumptions. It is well known that a similar statement is false if \(T_1\) and \(T_2\) are only diffeomorphisms. The authors' result does not depend on the Diophantine properties of the rotation number, and indeed holds for all irrational rotation numbers. The result is proved using renormalization methods. The main idea is that two maps with the same irrational rotation number and with the same local structure of their singular points (if there are any) belong to the same stable manifold for the renormalization operator. So two sequences of renormalizations created from these two maps approach one another at exponential rate. The particular notion of renormalization employed in this paper depends upon expansion of the rotation number as a continued fraction.