The multipliers of periodic points in one-dimensional dynamics

It will be shown that the smooth conjugacy class of an S-unimodal map which has neither a periodic attractor nor a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M Shub and D Sullivan (1985 Expanding endomorphism of the circle revisited Ergod. Theor. Dynam. Sys. 5 285-9) for smooth expanding maps of the circle.