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 in 1999. They have multiple applications in circle dynamics, with the case $s=-1$ being particularly important for describing conjugacy classes. In arxiv:2306.13524, 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.