Beau bounds for multicritical circle maps

Let f:S1→S1 be a C3 homeomorphism without periodic points having a finite number of critical points of power-law type. In this paper we establish real a-priori bounds, on the geometry of orbits of f, which are beau in the sense of Sullivan, i.e. bounds that are asymptotically universal at small scales. The proof of the beau bounds presented here is an adaptation, to the multicritical setting, of the one given by the second author and de Melo in de Faria and de Melo (1999), for the case of a single critical point.