Authomorphic measures with negative exponents for multicritical circle maps

Authomorphic or $ s $-measures for circle diffeomorphisms were introduced by R. Douady and J.-C. Yoccoz [1]. They have multiple applications in circle dynamics, with the case $ s = -1 $ being particularly important for describing conjugacy classes. In [5], E. de Faria, P. Guarino and B. Nussenzveig proved existence and uniqueness of automorphic $ s $-measures for multicritical circle maps for all $ s>0 $. The purpose of this paper is to extend these results to $ s $-measures with negative values of $ s $. As an application, we prove a smoothness result for irrational Arnold tongues in families of multicritical maps.