The root extraction problem

The Nth root extraction problem for germs of diffeomorphisms is the problem of finding a germ of diffeomorphism such that gN=f, where gN is the Nth iterate of g under composition. Depending on f and on the multiplier of g at the origin there can be formal and analytic obstructions to a solution of the problem. By considering an unfolding of f we explain these obstructions. Indeed each analytic obstruction corresponds to an accumulation of periodic points which, in turn, are an obstruction to taking an Nth root of the unfolding. We apply this to the problem of the section of a curvilinear angle in N equal parts in conformal geometry.