How violently do smooth iterative roots oscillate?

A crisis in the smoothness of iterative roots appears at fixed points. The local Cr smoothness of iterative roots near a hyperbolic fixed point is well known, but its global Cr extension to both fixed points is generically impossible because of their violent oscillation near the next hyperbolic fixed point. In this study, we consider how violently these smooth iterative roots oscillate. We prove that their first order derivatives are bounded but their r-th (r≥2) order derivatives are generically unbounded. In addition, we prove that the global C1 smoothness is of vital significance for the global higher order derivatives. As a corollary, we show that iterative roots on the maximal interval oscillate with C1 boundedness but Cr (r≥2) unboundedness.