The root extraction problem

Abstract

The N th root extraction problem for germs of diffeomorphisms f : ( C , 0 ) → ( C , 0 ) is the problem of finding a germ of diffeomorphism g : ( C , 0 ) → ( C , 0 ) such that g N = f , where g N is the N th 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 N th root of the unfolding. We apply this to the problem of the section of a curvilinear angle in N equal parts in conformal geometry.