12 - CIRCLE DIFFEOMORPHISMS

Summary

It turns out that adding sufficient differentiability to the ingredients of the previous chapter produces several new facets in the theory of circle maps. At the end of Section 11.2b we outlined a complete topological classification of circle homeomorphisms with irrational rotation number. When we restrict attention to sufficiently smooth diffeomorphisms (Theorem 12.1.1) the situation changes dramatically. The example of Proposition 12.2.1 shows that the smoothness required is almost sharp. The rotation number becomes a complete invariant of topological conjugacy. This is not dissimilar to the situation with hyperbolic dynamical systems (cf., for example, Theorems 2.6.1 and 2.6.3). On the other hand, the classification of circle diffeomorphism up to differentiable conjugacy is possible only for rotation numbers satisfying extra arithmetic conditions. In Section 12.3 we prove a local result of this kind in the analytic setting, while in Sections 12.5 and 12.6 we show that without an arithmetic condition a variety of pathological behaviors of the conjugacy may be produced at will. Finally we show in Section 12.7 that a certain aspect of the behavior of an irrational rotation, namely, ergodicity with respect to Lebesgue measure, is preserved for all sufficiently smooth circle diffeomorphisms.