Abstract: 

In the study of circle dynamics, the regularity of a map famously dictates its dynamical behavior—a theme dating back to Denjoy's celebrated C^1 vs. C^2 dichotomy. When considering monotone one-parameter families of circle diffeomorphisms, a natural question arises: how does the regularity of the family govern the regularity of the rotation number as a function of the family parameter? It is known that C^2 families yield a Hölder continuous rotation number with exponent at least 1/2 (Graczyk) and C^1 families guarantee at least log-Hölder continuity (Gorodetski & Kleptsyn). But is this the best we can do for generic C^2 families? And for a generic C^1 family, could Hölder continuity somehow survive? What happens with C^{1+\alpha} families? In this talk, we give partial answers to these questions by establishing a modern analogue to Denjoy's dichotomy: we show that for C2 families, the 1/2-Hölder continuity for the rotation number established by Graczyk is generically optimal, whereas for C1 families, Hölder continuity of the rotation number is generically lost entirely. We also comment on the behavior of families coming from a certain dense subset of the C^{1+\alpha} families.