The C1 Brunovsky Hypothesis

We prove that for the generic C1 arc of circle diffeomorphisms, t ft, the rotation number t ρ(ft) is a piecewise monotone function of t. A proof of this was attempted by Brunovsky in 1970. We also show that our methods cannot hope to answer the corresponding question for Cr arcs of circle diffeomorphisms when r is ≥ 2.